Mock modular Mathieu moonshine modules...Mock modular Mathieu moonshine modules Miranda C N Cheng1*,...

Mock modular Mathieu moonshine modulesMiranda C N Cheng1*, Xi Dong2, John F R Duncan3, Sarah Harrison2, Shamit Kachru2 and Timm Wrase2

1 IntroductionThe investigation of moonshine connecting modular objects, sporadic groups, and 2d conformal field theories has been revitalized in recent years by the discovery of several new classes of examples. While monstrous moonshine [3, 15, 21, 30, 36, 37, 44] remains the best understood and prototypical case, a new class of umbral moonshines tying mock modular forms to automorphism groups of Niemeier lattices has recently been uncovered [13, 14] (cf. also [6]). The best studied example, and the first to be discovered, involves the group M24 and was discovered through the study of the elliptic genus of K3 [35]. The twining functions have been constructed in [4, 31, 38, 39] and were proved to be the graded characters of an infinite-dimensional M24-module in [43]. Steps towards a better and deeper understanding of this mock modular moonshine can be found in [7, 8, 18, 40–42, 46, 56, 57, 59–61], and particularly in [9], where the importance of K3 surface geometry for all cases of umbral moonshine is elucidated. Possible connections to space-time physics in string theory have been discussed in [5, 47–49, 63]. Evidence for a deep connection between monstrous and umbral moonshine has appeared in [25, 55].

Abstract

We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distin-guished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co0 that fixes a 3-dimensional subspace of its unique non-trivial 24-dimen-sional representation commutes with a certain N = 4 superconformal algebra. Simi-larly, any subgroup of Co0 that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N = 2 superconformal algebra. Through the decomposition of the corresponding twined partition functions into characters of the N = 4 (resp. N = 2) superconformal algebra, we arrive at mock modular forms which coincide with the graded characters of an infinite-dimensional Z-graded module for the corresponding group. The Mathieu groups are singled out amongst various other possibilities by the moonshine property: requiring the corresponding weak Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions constitute the first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman–Sims, are also discussed.

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RESEARCH

Cheng et al. Mathematical Sciences (2015) 2:13 DOI 10.1186/s40687-015-0034-9

*Correspondence: [email protected] 1 Institute of Physics and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The NetherlandsFull list of author information is available at the end of the article

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In none of these cases, however, has a connection to an underlying conformal field theorya (whose Hilbert space furnishes the underlying module) been established. The goal of this paper is to provide first examples of mock modular moonshine for sporadic simple groups G, where the underlying G-module can be explicitly constructed in the state space of a simple and soluble conformal field theory.

Our starting point is the Conway module sketched in [36], studied in detail in [24], and revisited recently in [27]. The original construction was in terms of a supersymmetric theory of bosons on the E8 root lattice, but this has the drawback of obscuring the true symmetries of the model. In [24], a different formulation of the same theory, as a Z2 orbifold of the theory of 24 free chiral fermions, was introduced. A priori, the theory has a Spin(24) symmetry. However, one can also view this theory as an N = 1 supercon-formal field theory. The choice of N = 1 structure breaks the Spin(24) symmetry to a subgroup. In [24] it was shown that the subgroup preserving the natural choice of N = 1 structure is precisely the Conway group Co0, a double cover of the sporadic group Co1. In [24] it was shown that this action can be used to attach a normalized principal modu-lus (i.e. normalized Hauptmodul) for a genus zero group to every element of Co0.

In this paper, we show that generalizations of the basic strategy of [24, 27] can be used to construct a wide variety of new examples of mock modular moonshine. Instead of choosing an N = 1 superconformal structure, we choose larger extended chiral alge-bras A. The subgroup of Spin(24) that commutes with a given choice can be determined by simple geometric considerations; in the cases of interest to us, it will be a subgroup that preserves point-wise a 2-plane or a 3-plane in the 24 dimensional representation of Co0, or equivalently, a subgroup that acts trivially on two or three of the free fermions in some basis. In the rest of the paper, we will use 24 to denote the unique non-trivial 24-dimensional representation of Co0, and use the term n-plane to refer to a n-dimen-sional subspace in 24.

It is natural to ask about the role in moonshine, or geometry, of n-planes in 24 for other values of n. One of the inspirations for our analysis here is the recent result of Gab-erdiel–Hohenegger–Volpato [40] which indicates the importance of 4-planes in 24 for non-linear K3 sigma models. The relationship between their results and the Co0-module considered here is studied in [25], where connections to umbral moonshine for various higher n are also established. We refer the reader to Sect. 9, or the recent articles [1, 10], for a discussion of the interesting case that n = 1.

In this work we will focus on the cases where A is an N = 4 or N = 2 superconformal algebra, though other possibilities exist. In the first case, we demonstrate that any sub-group of Co0 that preserves a 3-plane in the 24-dimensional representation can stabilise an N = 4 structure. The groups that arise are discussed in e.g. Chapter 10 of the book by Conway and Sloane [16]. They include in particular the Mathieu groups M22 and M11. In the second case, where the group need only fix a 2-plane in order to preserve an N = 2 superconformal algebra, there are again many possibilities (again, see [16]), including the larger Mathieu groups M23 and M12. Note that the larger the superconformal algebra we wish to preserve, the smaller the global symmetry group is. Corresponding to cer-tain specific choices of the N = 0, 1, 2, 4 algebras, we have the global symmetry groups Spin(24) ⊃ Co0 ⊃ M23 ⊃ M22.


We should stress again that there are other Co0 subgroups that preserve N = 4 resp. N = 2 superconformal algebras arising from 3-planes resp. 2-planes in 24. Some exam-ples are: the group U4(3) for the former case, and the McLaughlin (McL) and Hig-man–Sims (HS) sporadic groups, and also U6(2) for the latter case. However, only for the Mathieu groups do the twined partition functions of the module display uniformly a special property, which we regard as an essential feature of the moonshine phenomena. Namely, all the mock modular forms obtained via twining by elements of the Mathieu groups are encoded in Jacobi forms that are constant in the elliptic variable, in the limit as the modular variable tends to any cusp other than the infinite one. This property also holds for the Jacobi forms of the Mathieu moonshine mentioned above, and may be regarded as a counter-part to the genus zero property of monstrous moonshine, as we explain in more detail in Sect. 8.

The importance of this property is its predictive power: it allows us to write down trace functions for the actions of Mathieu group elements with little more information than a certain fixed multiplier system, and the levels of the functions we expect to find. A priori these are just guesses, but the constructions we present here verify their validity.

This may be compared to the predictive power of the genus zero property of monstrous moonshine: if Ŵ < SL2(R) determines a genus zero quotient of the upper-half plane, and if the stabilizer of i∞ in Ŵ is generated by ±

(1 10 1

), then there is a unique Ŵ-invariant hol-

omorphic function satisfying TŴ(τ) = q−1 + O(q) as τ → i∞, for q = e2π iτ . The miracle of monstrous moonshine, and the content of the moonshine conjectures of Conway–Norton [15], is that for suitable choices of Ŵ, the function TŴ is the trace of an element of the monster on some graded infinite-dimensional module (namely, the moonshine module of [36, 37]). The optimal growth property formulated in [14] plays the analogous predictive role in umbral moonshine, and is similar to the special property we formulate for the Mathieu moonshine considered here.

We mention here that although the moonshine conjectures have been proven in the monstrous case by Borcherds [3], and verified in [7, 28, 43] (see also [6]) for umbral moonshine, conceptual explanations of the genus zero property of monstrous moon-shine, and of the analogous properties of umbral moonshine, and the Mathieu moon-shine studied here, remain to be determined. An approach to establishing the genus zero property of monstrous moonshine via quantum gravity is discussed in [29].

The organization of the paper is as follows. We begin in Sect. 2 with a review of the module discussed in [27]. In Sect. 3, we describe methods for endowing this module with N = 4 and N = 2 structure. In Sect. 4, we discuss what this does to the mani-fest symmetry group of the model, reducing the symmetry from Co0 to a variety of other possible groups which preserve a 3-plane (respectively 2-plane) in the 24 of Co0. In Sects. 4 and 5, we discuss the action of these Co0-subgroups on the mod-ules and compute the corresponding twining functions. We identify M23, M22, McL, HS, U6(2) and U4(3) as some of the most interesting Co0-subgroups preserving some extended superconformal algebra. In Sects. 6 and 7, we discuss in some detail the decomposition of the graded partition function of our chiral conformal field theory into characters of irreducible representations of the N = 4 and N = 2 superconfor-mal algebras. In Sect. 8, we discuss the special property we require from a moon-shine twining function, and show how this property singles out the Mathieu groups


in our setup. We close with a discussion in Sect. 9. The appendices contain a number of tables: character tables for the various groups we discuss, tables of coefficients of the vector-valued mock modular forms that arise as our twining functions, and tables describing the decompositions of our modules into irreducible representations of the various groups.

2 The free field theoryThe chiral 2d conformal field theory that will play a starring role in this paper has two different constructions. The first is described in [37] and starts with 8 free bosons Xi compactified on the 8-dimensional torus given by the E8 root lattice, together with their Fermi superpartners ψ i. One then orbifolds by the Z2 symmetry

Note that, in more mathematical terms, a chiral 2d conformal field theory can be under-stood to mean a super vertex operator algebra (usually assumed to be simple, and of CFT-type in the sense of [23]), together with a (simple) canonically-twisted module. These two spaces are referred to as the Neveu–Schwarz (NS) and Ramond (R) sectors of the theory, respectively. The compactification of free bosons on the torus defined by a lattice L manifests, in the NS sector, as the usual lattice vertex algebra construction, and their Fermi superpartners are then realized by a Clifford module, or free fermion, super vertex algebra, where the underlying orthogonal space comes equipped with an isometric embedding of L. In the cases under consideration, there is a unique simple canonically-twisted module up to equivalence (cf. e.g. [24]), and hence a unique choice of R sector.

The orbifold procedure is described in detail in the language of vertex algebra in [24]. In what follows, a field of dimension d is a vertex operator or intertwining operator attached to a vector v in the NS or R sector, respectively, with L(0)v = dv. A current is a field of dimension 1. A field is called primary if its corresponding vector is a highest weight for the Virasoro (Lie) algebra. A ground state is an L(0)-eigenvector of minimal eigenvalue.

The E8 construction just described has manifest N = 1 supersymmetry, in the sense that the Neveu–Schwarz and Ramond algebras act naturally on the NS and R sectors, respectively. After orbifolding we obtain a c = 12 theory with no primary fields of dimension 12. The partition functions (i.e., graded dimensions) of this free field theory can easily be computed. For example, the NS sector partition function is given by

where E4 is the weight 4 Eisenstein series, being the theta series of the E8 lattice, η(τ) = q1/24

∏∞n=1(1− qn) is the Dedekind eta function, and θi are the Jacobi theta func-

tions recorded in “Appendix A”. We have also set q = ǫ(τ ) and we use the shorthand notation e(x) = e2π ix throughout this paper.

(2.1)(Xi,ψ i) → (−Xi,−ψ i).

(2.2)ZNS,E8(τ ) =1

2

(E4(τ )θ3(τ , 0)

4

η(τ)12+ 16

θ4(τ , 0)4

θ2(τ , 0)4+ 16

θ2(τ , 0)4

θ4(τ , 0)4

)

(2.3)= q−1/2 + 0+ 276q1/2 + 2048q + 11202q3/2 + · · · ,


One recognizes representations of the Co1 sporadic group appearing in the q-series (2.3): apart from 276, which is the minimal dimension of a faithful irreducible represen-tation (cf. [31]), one can also observe

In fact, this theory has a Co0 ∼= 2.Co1 symmetry, which we call non-manifest since the action of Co0 is not obvious from the given description. Note that we sometimes use n or Zn to denote Z/nZ depending on the context.

A better realization, for our purposes, was discussed in detail in [28] (cf. also [40]). The E8 orbifold theory is equivalent to a theory of 24 free chiral fermions �1, �2, . . . , �24, also orbifolded by the Z2 symmetry �α → −�α. This gives an alternative description of the Conway module above. The partition function from this “free fermion” point of view is more naturally written as

This is equal to (2.2) according to non-trivial identities satisfied by theta functions. Note that θ1(τ , 0) = 0.

The free fermion theory has a manifest Spin(24) symmetry, but not a manifest N = 1 supersymmetry. However, one can construct an N = 1 supercurrent as follows. There is a unique (up to scale) NS ground state, but there are 212 = 4096 linearly independent Ramond sector ground states, which may be obtained by acting on a given fixed R sector ground state with the fermion zero modes �i(0). It will be convenient to label the result-ing 4096 Ramond sector ground states by vectors s ∈ F

122 , where F2 = {−1/2, 1/2}.

We therefore have 4096 spin fields of dimension 32 which implement the flow from the NS to the R sector. Denoting these fields as Ws, one can try to find a linear combination

which will serve as an N = 1 supercurrent (i.e., field whose modes generate actions of the Neveu–Schwarz and Ramond super Lie algebras). As demonstrated in [28], and as we will review in the next section, there exists a set of values cs such that the operator product expansion of W and the stress tensor T close properly, defining actions of the Neveu–Schwarz and Ramond algebras.

Any choice of W breaks the Spin(24) symmetry, since the Ramond sector ground states split into two 2048-dimensional irreducible representation of Spin(24). It is proven in [28] that the subgroup of Spin(24) that stabilizes a suitably chosen N = 1 supercurrent is exactly the Conway group Co0. In brief, the method of [28] is to identify a certain ele-mentary abelian subgroup of order 212 in Spin(24) (which should be regarded as a copy of the extended Golay code in Spin(24)). The action of this subgroup on the Ramond sector ground states singles out a particular choice of W, with the property that it is not

(2.4)2048 = 1+ 276+ 1771,

(2.5)11202 = 1+ 276+ 299+ 1771+ 8855,

· · ·

(2.6)ZNS,fermion(τ ) =1

2

4∑

i=2

θ12i (τ , 0)

η12(τ ).

(2.7)W =

∑

s∈F122

csWs


annihilated by the zero mode of any dimension 1 field in the theory. It follows from this (cf. Proposition 4.8 of [28]) that the subgroup of Spin(24) that stabilizes W is a reductive algebraic group of dimension 0, and hence finite. On the other hand, one can show (cf. Proposition 4.7 of [28]) that this group contains Co0, by virtue of the choice of subgroup 212. We obtain that the full stabilizer of W in Spin(24) is Co0 by verifying (cf. Proposition 4.9 of [28]) that Co0 is a maximal subgroup, subject to being finite.

In the rest of this paper, we extend this idea as follows. Instead of choosing an N = 1 supercurrent and viewing the theory as an N = 1 super conformal field theory, we choose various other super extensions of the Virasoro algebra. We will argue that N = 4 and N = 2 superconformal presentations of the theory are in one to one correspond-ence with choices of subgroups of Co0 which fix a 3-plane (respectively, 2-plane) in the 24 dimensional representation. This leads us naturally to theories with various inter-esting symmetry groups, whose twining functions are easily computed in terms of the partition function (or elliptic genus) of the free fermion conformal field theory. These functions in turn are expressed nicely in terms of mock modular forms, and thus we establish mock modular moonshine relations for subgroups of Co0 via this family of modules.

3 The superconformal algebrasWe first discuss the largest superconformal algebra (SCA) we will consider, which gives rise to smaller global symmetry groups. We will construct an N = 4 SCA in the free fermion orbifold theory. Our strategy is to first construct the SU(2) fields, and act with them on an N = 1 supercurrent to generate the full N = 4 SCA. We consequently obtain actions of the N = 2 SCA by virtue of its embeddings in the N = 4 SCA. In this process, we break the Co0 symmetry group down to a proper subgroup as we will discuss in Sect. 4. We refer the reader to [32, 50, 52] for background on the N = 4 and N = 2 superconformal algebras.

We start with 24 real free fermions �1, �2, . . . , �24 . Picking out the first three fermions, we obtain the currents Ji:

They form an affine SU(2) algebra with level 2 as may be seen from their operator prod-uct expansion (OPE),

The next step is to pick an N = 1 supercurrent and act with Ji on it. As we reviewed in Sect. 2, an N = 1 supercurrent exists in this model and may be written as a linear com-bination of spin fields. Moreover, it may be chosen so that its stabilizer in Spin(24) is precisely Co0. We will present a very general version of the construction now, and then extend it to find the N = 4 SCA.

To write the N = 1 supercurrent explicitly, we first group the 24 real fermions into 12 complex ones and bosonize them:

(3.1)Ji = −iǫijk�j�k , i, j, k ∈ {1, 2, 3} .

(3.2)Ji(z)Jj(0) ∼1

z2δij +

i

zǫijk Jk(0) .


In terms of the bosonic fields H = (H1, . . . ,H12), an N = 1 supercurrent W may be writ-ten as

where each component of s = (s1, s2, . . . , s12) takes the values ±1/2, and the coefficients ws belong to C. We have introduced cocycle operators cs(p) to ensure that the fields with integer spins (i.e., corresponding to even parity vectors) commute with all other operators, and the fields with half integral spins (corresponding to odd parity vectors) anticommute amongst themselves. The cocycle operators depend on the zero-mode operators p which are characterized by the commutation relation

where k = (k1, . . . , k12) is an arbitrary 12-tuple of complex numbers.The associativity and closure of the OPE of the “dressed” vertex operators

requires that

where the ǫ(k, k′) satisfy the 2-cocycle condition

Moreover, in order for Vk to have the desired (anti)commutation relation, the condition

should be imposed. An explicit description of the cocycle for a general vertex operator eik·H may be chosen as

according to [33]. In our case, M is a 12× 12 matrix that has the block form

where 14 is the 4 × 4 matrix with all entries set to 1.Generically, the OPE of W with itself is

(3.3)

ψa ≡ 2−1/2(�2a−1 + i�2a) ∼= e

iHa ,

ψa ≡ 2−1/2(�2a−1 − i�2a) ∼= e

−iHa , a = 1, 2, . . . , 12.

(3.4)W =

∑

s∈F122

wseis·Hcs(p),

(3.5)[p, eik·H

]= keik·H,

(3.6)Vk = eik·Hck(p)

(3.7)ck(p+ k′)ck′(p) = e(k, k′)ck+k′(p),

(3.8)e(k, k′) e(k + k′, k

′′) = e(k′, k′′) ǫ(k, k′ + k′′).

(3.9)e(k, k′) = (−1)k·k′+k2k′2e(k′, k)

(3.10)ck(p) = eiπk·M·p

(3.11)M =

M4 0 014 M4 014 14 M4

, M4 =

0 0 0 01 0 0 01 1 0 0−1 1 − 1 0

,

(3.12)W (z)W (0) ∼ ww

[1

z3+ T (0)

4z

]+ wŴαβγ δw

96z�α�β�γ �δ(0),


where we have defined

for α < β < γ < δ. The other components of Ŵαβγ δ are defined by the requirement that it is totally antisymmetrized. For W to be an N = 1 supercurrent, the last terms in (3.12) must vanish,

and the first two terms must have the correct normalization,

From now on we presume to be chosen a solution (ws) such that W is an N = 1 super-current stabilized by Co0, as described in Sect. 2.

We may now act with the SU(2) currents Ji on our N = 1 supercurrent W. In order to do this, write the SU(2) currents in (3.1) in bosonized form,

Here we have included the cocycles e±iπp1. We now extract the singular terms of the OPEs,

where Wi are slightly modified combinations of spin fields,

and where Rs ≡ (−s1,−s2, s3, . . . , s12).

(3.13)ws = w−sc−s(s),

(3.14)

Ŵαβγ δ

ss′ =(ŴαŴβŴγ Ŵδ

)ss′c−s(s

′ − s)

(−2s⌈ α

2⌉+1

)· · ·

(−2s⌈ β

2⌉

)(−2s⌈ γ

2⌉+1

)· · ·

(−2s⌈ δ

2⌉

),

(3.15)wŴαβγ δw = 0 , ∀α,β , γ , δ ∈ {1, 2, . . . , 12},

(3.16)ww =

∑

s∈F122

w−swscs(−s) = 8.

(3.17)J1 = −1

2

(eiH1 − e−iH1

)(eiH2eiπp1 + e−iH2e−iπp1

),

(3.18)J2 =i

2

(eiH1 + e−iH1

)(eiH2eiπp1 + e−iH2e−iπp1

),

(3.19)J3 = i∂H1.

(3.20)Ji(z)W (0) ∼ − i

2zWi(0),

(3.21)W1 = −∑

s

2s2wRseis·Hcs(p),

(3.22)W2 = i∑

s

4s1s2wRseis·Hcs(p),

(3.23)W3 = i∑

s

2s1wseis·Hcs(p),


We claim that all three Wi defined above are valid N = 1 supercurrents. This is because we may obtain, for instance, W3 from W by rotating the 1–2 plane by π, and the conditions (3.15) and (3.16), for being an N = 1 supercurrent, are invariant under SO(24) rotations. We may obtain W2 and W3 similarly. This shows that each of the Wi is an N = 1 supercurrent.

Furthermore, we can check using the identity (3.15) that the OPEs of the Wi are given by

This shows that W, Wi, and Ji, together with the stress tensor T, defined as

form an N = 4 SCA with central charge c = 12. We may recombine the four N = 1 supercurrents W, Wi into the more conventional N = 4 supercurrents

which transform according to the representation 2+ 2 of SU(2). In terms of these super-currents we obtain the standard (small) N = 4 SCA with central charge c = 12, charac-terized by the following set of OPEs:

(3.24)Wi(z)Wj(0) ∼ δij

[8

z3+ 2

zT (0)

]+ 2iǫijk

[2

z2Jk(0)+

1

z∂Jk(0)

],

(3.25)W (z)Wi(0) ∼ −2i

(2

z2+ ∂

z

)Ji(0),

(3.26)Ji(z)Wj(0) ∼i

2z(δijW + ǫijkWk).

(3.27)T = −1

2

∑

α

�α∂�α = −1

2

∑

a

∂Ha∂Ha,

(3.28)W±1 ≡ 2−1/2(W ± iW3) , W±

2 ≡ ±2−1/2i(W1 ± iW2),

(3.29)T (z)T (0) ∼ 6

z4+ 2

z2T (0)+ 1

z∂T (0),

(3.30)T (z)W±a (0) ∼ 3

2z2W±

a (0)+ 1

z∂W±

a (0),

(3.31)T (z)Ji(0) ∼1

z2Ji(0)+

1

z∂Ji(0),

(3.32)W+a (z)W−

b (0) ∼ δab

[8

z3+ 2

zT (0)

]− 2σ i

ab

[2

z2Ji(0)+

1

z∂Ji(0)

],

(3.33)W+a (z)W+

b (0) ∼ W−a (z)W−

b (0) ∼ 0,

(3.34)Ji(z)W+a (0) ∼ − 1

2zσ iabW

+b (0),

(3.35)Ji(z)W

−a (0) ∼ 1

2zσ i∗abW

−b (0),


Here σ i are the Pauli matrices.Now we can generalize our formula for the partition function (2.6) to include a grading

by the U(1) charge under the Cartan generator of the SU(2). The U(1) charge operator J0 is, by definition, twice the zero-mode of the J3 current. From this and the definition J3 = −i�1�2 = ψ1ψ1, we see that under J0 the complex fermion ψ1 has charge 2 while the other 11 complex fermions are neutral. Therefore, the U(1)-graded NS sector partition function becomes

In the above discussion, we have chosen the first three fermions out of a total of 24 to generate a set of SU(2) currents. Together with an N = 1 supercurrent they gen-erate a full N = 4 SCA. It is clear that we are free to choose any three fermions for this purpose. In fact, we could choose an arbitrary three-dimensional subspace of the 24-dimensional vector space spanned by the fermions, and obtain an N = 4 SCA. For a given N = 1 supercurrent, not all choices of 3-plane are equivalent, as we will see in Sect. 4.

Observe that we could instead have chosen to single out only two real fermions, and construct a U(1) current algebra instead of an SU(2) current algebra. Completely analo-gous manipulations then show that each such choice provides an N = 2 superconformal algebra. As a result we can equip the Co0 theory with N = 2 structure in such a way that the global symmetry group is broken to subgroups G of Co0 which stabilize 2-planes in 24.

To summarize the results of this section, we have shown how to construct an N = 1 supercurrent for the chiral conformal field theory described, in the previous section, as an orbifold of 24 free fermions. We have also shown how choices of 2- and 3-planes in the space spanned by the generating fermions give rise to actions of the N = 2 and N = 4 superconformal algebras (respectively) on the theory. As reviewed in Sect. 2, a suitable choice of N = 1 structure reduces the global symmetry of the theory to Co0. In the next section we will discuss the finite simple groups that appear when we impose the richer, N = 2 and N = 4 superconformal structures.

4 Global symmetriesEnhancing the N = 1 structure of the theory to N = 4 breaks the Co0 symmetry. We now show that for a specific choice of 3-plane in 24, resulting in a specific copy of the N = 4 SCA, the stabilising subgroup of Co0 is the sporadic group M22. Similarly, for a specific choice of 2-plane, resulting in a specific copy of the N = 2 SCA, the sta-bilising subgroup of Co0 is the sporadic group M23. This amounts to a proof that the model described in Sect. 2 results in an infinite-dimensional M22 (resp. M23)-module underlying the mock modular forms described in Sect. 6 (resp. Sect. 7) arising from its

(3.36)Ji(z)Jj(0) ∼1

z2δij +

i

zǫijk Jk(0).

(3.37)ZNS(τ , z) =1

2

4∑

i=2

θi(τ , 2z) θi(τ , 0)11

η(τ)12.


interpretation as an N = 4 (resp. N = 2) module. More generally, we establish the mod-ules for 3- (2-)plane-fixing subgroups of the largest Mathieu group M24 by fixing a spe-cific copy of N = 4 (N = 2) SCA.

Recall that the theory regarded as an N = 0 theory (i.e., with no extension of the Virasoro action) has a Spin(24) symmetry resulting from the SO(24) rotations on the 24-dimensional space, and a suitable choice of N = 1 supercurrent breaks the Spin(24) group down to its subgroup Co0. The group Co0 is the automorphism group of the Leech lattice �Leech, and various interesting subgroups of Co0 can be identified as stabilizers of suitably chosen lattice vectors in �Leech. To study the automorphism group of the mod-ule when fixing more structure—more supersymmetries in this case—it will therefore be useful to describe the enhanced supersymmetries in terms of Leech lattice vectors.

In Chapter 10 of [16] it is shown that if we choose an appropriate tetrahedron in the Leech lattice, whose edges have lengths squared 16× (2, 2, 2, 2, 3, 3) in the normali-sation described below, the subgroup of Co0 that leaves all vertices of the tetrahedron invariant is M22. To be more precise, let eγ, for γ ∈ {1, 2, . . . , 24}, be an orthonormal basis of R24, and choose a copy G of the extended binary Golay code in P({1, . . . , 24}). Then we may realize �Leech as the lattice generated by the vectors 2

∑γ∈C eγ for

C ∈ G together with −4e1 +∑24

γ=1 eγ. (One can show that all 24 vectors of the form −4eα +

∑24γ=1 eγ are in �Leech.) Define the tetrahedron T{α,β} to be that whose four verti-

ces are O = 0, Xα = 4eα +∑24

γ=1 eγ, Xβ = 4eβ +∑24

γ=1 eγ and Pαβ = 4eα + 4eβ, for any α,β ∈ {1, 2, . . . , 24} with α �= β. For every such T{α,β}, the subgroup fixing every vertex is a copy of M22, a sporadic simple group of order 27 · 32 · 5 · 7 · 11 = 443, 520 and the subgroup of M24 fixing eα and eβ.

From the above discussion, it is clear that given {α,β}, a copy of M22 stabilises the real span of eα, eβ and

∑24γ=1 eγ. Given a suitable choice of the N = 1 superconformal alge-

bra, this copy of R3 in 24 then determines, up to rotations, the three fermions, denoted �1,2,3, from which the SU(2) current algebra was built in Sect. 3. By definition then, a copy of M22 leaves the N = 4 superconformal algebra invariant.

A natural question is: what is the symmetry group G that fixes a given choice of N = 2 superconformal structure? Given the above description of the M22 action, we can choose the R2 ⊂ R3 generated by eα and

∑24γ=1 eγ and use the two free fermions lying in the R2

to construct the N = 2 sub-algebra of the N = 4 SCA. Specifically, the U(1) action is rotation of the R2. From the above discussion, it is not hard to see that there is a copy of M23 fixing eα and

∑24γ=1 eγ and hence stabilising the N = 2 structure. Recall that M23

is a sporadic simple group of order 27 · 32 · 5 · 7 · 11 · 23 = 10, 200, 960. In terms of the Leech lattice, it corresponds to the fact that the stabiliser of the triangle in �Leech whose edges have lengths squared 16× (6, 3, 2), with vertices chosen to be O, Xα and 2

∑24γ=1 eγ

, is a copy of M23 inside the copy of Co0 stabilising �Leech.This furnishes a proof that the theory described in Sect. 2 leads to modules for M22

and M23 which explicitly realize the mock modular forms to be defined in Sects. 6 and 7.We should mention that by stabilizing different choices of geometric structure, other

than the tetrahedron and triangle just discussed, leading to M22 and M23, respectively, we can determine other global symmetry groups G. Indeed, our method constructs a G-module with N = 4 (N = 2) superconformal symmetry for any subgroup G < Co0


which fixes a 3-plane (2-plane) in 24. Since, as we will see in Sect. 6 (Sect. 7), such mod-ules furnish assignments of mock modular forms to the elements of their global symme-try groups, it is an interesting question to classify the global symmetry groups G < Co0 that can arise. We conclude this section with a discussion of some of these possibilities. Certainly a full classification is beyond the scope of this work, so we restrict our atten-tion (mostly) to sporadic simple examples.

Indeed, the Conway group Co0 is a rich source of sporadic simple groups, for no less than 12 of the 26 sporadic simple groups are involved in Co0 (cf. [17]), in the sense that they may be obtained by taking quotients of subgroups of Co0. Of these 12, all but 3 are actually realised as subgroups, and 6 of these 9 sporadic simple groups appear as sub-groups of Co0 fixing (at least) a 2-plane in 24. These six 2-plane fixing groups are the smaller Mathieu groups, M23, M22, M12 and M11, the Higman–Sims group HS, and the McLaughlin group McL. Some 2-planes they fix are described explicitly in Chapter 10 of [16].N = 4 modulesFrom the character tables (cf. [17]) of the six sporadic 2-plane fixing subgroups of Co0

it is clear that M22 and M11 are the only examples that fix a 3-plane. Even though M11 is not a subgroup of M22, it turns out that the mock modular forms attached to M11 by our N = 4 construction (and the analysis of Sect. 6) are a proper subset of those attached to M22, since the conjugacy classes of Co0 appearing in a 3-plane-fixing subgroup M11 are a proper subset of those appearing in a subgroup M22. For this reason we focus on M22 when discussing mock modular forms attached to sporadic simple groups via the N = 4 construction in this work.

If we expand our attention to simple, not necessarily sporadic subgroups of Co0, then there is one example which is larger than M22 (which has order 443,520). Namely, the group U4(3), with order 3,265,920, can arise as the stabilizer of a suitably chosen 3-plane in the 24 of Co0 [16]. The U4(3) characters are presented in Appendix Table 20, the coefficients in the associated (twined) vector valued mock modular forms in Appendix Tables 3 and 4, and the decomposition of the module into irreducible representations of the group in Appendix Tables 26 and 27.

As we shall see in Sect. 8, the Jacobi forms attached to M22 (and therefore also those attached to M11) by the N = 4 construction are distinguished in that they satisfy a nat-ural analogue of the genus zero condition of monstrous moonshine. By contrast, this property does not hold for all the Jacobi forms arising from U4(3). This is the main rea-son for our focus on M22 in the context of N = 4 supersymmetry (Appendix Tables 1, 2, 5, 6, 15).N = 2 modulesWe have focused on the example of M23, with order 10,200,960, in this section. Since

M22 and M11 are subgroups of M23 we do not consider them further in the context of N = 2 structures. Of the remaining sporadic simple 2-plane-fixing subgroups of Co0, the largest is the McLaughlin group McL, which is actually considerably larger than M23, having order 898,128,000. Its characters are presented in Appendix Table 21, the coefficients of the (twined) mock modular forms in Appendix Tables 7 and 8, and the decomposition of the module into irreducible representations of the group in Appendix Tables 33, 34, 35, 36, 37, 38.


The next largest example, also larger than M23, is the Higman–Sims group HS, with order 44,352,000. Its characters are presented in Appendix Table 22, the coefficients of the (twined) mock modular forms in Appendix Tables 9 and 10, and the decomposition of the module into irreducible representations of the group in Appendix Tables 39, 40, 41, 42, 43, 44.

If we expand our attention to simple groups fixing a 2-plane in 24 then there is one example larger than McL. Namely, the group U6(2), of order 9,196,830,720, fixes any tri-angle in 24 whose three sides are vectors of minimal length in the Leech lattice. The characters of U6(2) are given in Appendix Tables 17, 18, 19, the coefficients of the (twined) mock modular forms in Appendix Tables 11, 12, 13, 14, and the decomposition of the module into irreducible representations of the group in Appendix Tables 45, 46, 47, 48, 49, 50, 51, 52, 53, 54.

In direct analogy with the case of N = 4 structure, it will develop in Sect. 8 that the Jacobi forms attached to M12 and M23 satisfy a natural analogue of the genus zero condi-tion of monstrous moonshine, and, contrastingly, this property fails in general for the modular forms arising from the other, non-Mathieu, 2-plane-fixing simple groups men-tioned above. For these reasons, and since M12 is relatively small, we focus on M23 in our discussion of N = 2 supersymmetry.

5 Twining the moduleIn the last sections, we have described how to equip the orbifolded free fermion the-ory with N = 4 and N = 2 superconformal structures. In this section we will use the Ramond sector of our theory to attach two variable formal power series—the g-twined graded R sector partition function, cf. (5.10)—to each element g ∈ Co0 that preserves at least a 2-plane in 24.

Let us denote the Ramond sector by V, and let us choose a U(1) charge operator J0. This will be twice the Cartan generator of the SU(2) in the N = 4 case, or the single U(1) generator in the case of N = 2 SCA. Then it is natural to define the Ramond-sector U(1)-graded partition function, or elliptic genus,

where we have introduced a chemical potential for the J0 charges and set y = e(z) for z ∈ C. Also, we define (−1)F as an operator on V by requiring that it act as Id on the untwisted free fermion contribution to V, and as −Id on the twisted fermion contribution.

As is expected for 2d conformal field theories with N ≥ 2 supersymmetry, the elliptic genus (5.1) transforms as a Jacobi form of weight 0, index m = c

6 and level 1.

(5.1)Z(τ , z) = T rV (−1)FqL0−c/24yJ0

(5.2)= 1

2

1

η12(τ )

4∑

i=2

(−1)i+1θi(τ , 2z)θ11i (τ , 0)

(5.3)= 1

2

E4(τ )θ41 (τ , z)

η12(τ )+ 8

4∑

i=2

(θi(τ , z)

θi(τ , 0)

)4

,


Explicitly, and since c = 12 in our case, this means that Z|2(�,µ) = Z for all �,µ ∈ Z, and Z|0,2γ = Z for all γ ∈ SL2(Z), where the elliptic and modular slash operators are defined by

respectively, for �,µ ∈ Z and γ =(a bc d

)∈ SL2(Z). (A Jacobi form of level N is only

required to satisfy φ|k ,mγ = φ for γ in the congruence subgroup Ŵ0(N ) (cf. (6.24)).)As we have seen in Sect. 2, the two different ways of writing this function, (5.2) and

(5.3), are intuitively connected more closely with the free fermion and E8 root lattice descriptions of the theory, respectively. Of course, the U(1)-graded NS sector partition function (3.37) is related to the above, graded Ramond sector partition function by a spectral flow transformation

There is a natural way in which one can twine the above function under certain sub-groups of Co0. From the previous discussions, we see that the representation 24 plays a central role in the way various subgroups of Co0 act on the model. Let’s denote by ℓg ,k and ℓg ,k, for k = 1, . . . 12, the 12 complex conjugate pairs of eigenvalues of g ∈ Co0 when acting on 24. This information is conveniently encoded in the so-called Frame shape of g, given by

satisfying ∑

n Lnmn = 24, through the fact that the 12 pairs {ℓg ,k , ℓg ,k} are precisely the 24 roots solving the equation

As discussed in Sects. 3 and 4, in order to preserve at least N = 2 superconformal symmetry and hence be able to twine the graded R-sector partition function (5.1), the subgroup G must leave at least a 2-dimensional subspace in 24 pointwise invariant. In the graded partition function this corresponds to leaving the factor θi(τ , 2z) in (5.2) invariant. As a result, for every conjugacy class [g] of such a group G we can choose ℓg ,1 = ℓg ,1 = 1. It is easy to see that when acting on the untwisted free fermions of the theory, contributing the terms involving θi with i = 3, 4 in (5.2), the group element g sim-ply replaces θ11i (τ , 0) with

where e(ρg ,k) = ℓg ,k.

(5.4)(φ|m(�,µ))(τ , z) = e(m(�2τ + 2�z))φ(τ , z + �τ + µ),

(φ|k ,mγ )(τ , z) = e(−m cz2

cτ+d)(cτ + d)−kφ

(aτ+bcτ+d

, zcτ+d

),

(5.5)ZNS(τ , z) = q1/2y−2Z(τ , z − τ+12 ).

�g =∏

n

Lnmn , 1 ≤ L1 < L2 < L3 · · · , and mn ∈ Z,mn �= 0,

∏

n

(xLn − 1)mn = 0.

(5.6)12∏

k=2

θi(τ , ρg ,k)


When trying to do the same for the contribution from the twisted fermions, contribut-ing the term involving θ2 in (5.2), however, we see that the above simple consideration suffers from an ambiguity. This can be seen from the fact that θ2(τ , ρ) = −θ2(τ , ρ + 1), and hence the answer cannot be determined simply by looking at the g-eigenvalues on 24. This of course is a reflection of the fact that the global symmetry group, with no superconformal structure imposed, is Spin(24), which is a 2-fold cover of SO(24). As a result, to specify the twining of the twisted fermion contribution, we also need to know the action of G on the faithful 212-dimensional representation of Spin(24) spanned by Ramond sector ground states in the free fermion theory (cf. Sect. 2), henceforth denoted 4096, which decomposes as 4096 = 1+ 276+ 1771+ 24 + 2024 in terms of the irre-ducible representations of Co0.

Note that, according to the orbifold construction, just “half” of the Ramond sec-tor ground states in the free fermion theory will contribute to the Ramond sector V of the orbifold theory under consideration. In terms of the Co0 action, the two “halves” are 24 + 2024, where Co0 acts faithfully, and 1+ 276+ 1771, where the action factors through Co1 = Co0/2. In practice, both choices give rise to equivalent theories (i.e., iso-morphic super vertex operator algebras, cf. [24, 27]), but they are inequivalent as Co0-modules. For us, the ground states represented by 24 + 2024 lie in the R sector, V, and the 1 in 1+ 276+ 1771 represents the Co0-invariant N = 1 supercurrent in the NS sec-tor of our orbifold theory.

The above discussion serves to remind us that there is, really, a vanishing term

in (5.2), which, for certain g ∈ Co0, will make a non-vanishing contribution to the g-twined version of (5.1). It vanishes when g = e is the identity because the Ramond sec-tor ground states in the free fermion theory come in pairs with opposite eigenvalues for (−1)F. Moreover, exchanging the pair corresponds to complex conjugation ψa ↔ ψa, for a = 1, . . . , 12, of the complex fermions. Recall that one of the complex fermions, denoted ψ1 in (3.19), was used to construct the U(1) charge operator J0, and we are inter-ested in the graded partition function where we introduce a chemical potential z for this operator. Because exchanging ψ1 ↔ ψ1 also induces a flip of U(1) charges, captured by z ↔ −z, the contribution of the first complex fermion does not vanish, corresponding to the fact that the identity

only forces θ1(τ , z) to vanish at z ∈ Z. Consequently, the g-twining of (5.7) makes a non-zero contribution to the g-twining of (5.2) if and only if ρg ,k �∈ Z for all k = 2, . . . , 12. In other words, it is non-zero only when the cyclic group generated by g fixes nothing but a 2-plane.

By inspection we find that, among the groups we consider, such group elements must be in the conjugacy classes 23AB ⊂ M23, 6AB, 12AB, 12DE, 18AB ⊂ U6(2), 15AB, 30AB ⊂ McL, or 20AB ⊂HS. The pairs of these conjugacy classes corresponding to the letters A and B (or D and E) are mutually inverse, and so their respective traces,

(5.7)0 = 1

2

1

η12(τ )θ1(τ , 2z)θ

111 (τ , 0)

(5.8)θ1(τ , z) = θ1(τ , z + 2) = −θ1(τ ,−z)


on any representation, are related by complex conjugation. In terms of our construction, choosing one over the other is the same as choosing what one labels ψ1 and ψ1, and the same as choosing an orientation on the 2-plane fixed by the group element in 24. As a result, from (5.8) we see that the θ1 term in the partition functions twined by these con-jugate A (D) and B (E) classes come with an opposite sign.

Let us work with the principal branch of the logarithm, and choose ρg ,k ∈ [0, 1/2] in (5.6). Then, by direct computation—we must compute directly, for the choice of labels for mutually inverse conjugacy classes is not natural—we find that the signs in (5.10) are

and ǫg ,1 = −1 for the inverse classes, 23B ⊂ M23, 20B ⊂ HS,&c.Putting these different contributions together, we conclude that for every [g] ⊂ G

where G is a subgroup of Co0 preserving (at least) a 2-plane in 24, the corresponding g-twined U(1)-graded R sector partition function reads

where

and where the ǫg ,1 are as determined in the preceding paragraph.In this section we have introduced the g-twined U(1)-graded Ramond sector partition

function, or g-twined elliptic genus of our theory, Zg, for any g ∈ Co0 fixing a 2-plane in 24. We have also derived an explicit formula (5.10) for Zg, in terms of the Frame shapes �g and values T r4096g. This Frame shape and trace value data is collected, for g ∈ G, for various G ⊂ Co0, in “Appendix B”. In Sects. 6 and 7 we will see how the above twin-ing leads to the mock modular forms playing the role of the McKay–Thompson series in these new examples of mock modular moonshine.

6 The N = 4 decompositionsFrom the discussion in Sect. 3 it is clear that the orbifold theory discussed in Sect. 2 can be equipped with N = 4 superconformal structure. In this section we will study the decomposition of the Ramond sector V into irreducible representations of the N = 4

(5.9)lǫg ,1 = 1 for g in

23A ⊂ M23,20A ⊂ HS,15A ∪ 30A ⊂ McL,12A ∪ 12D ∪ 6B ∪ 18B ⊂ U6(2),

(5.10)Zg (τ , z) = T rV g(−1)FqL0−c/24yJ0

(5.11)= 1

2

1

η(τ)12

4∑

i=1

(−1)i+1ǫg ,i θi(τ , 2z)

12∏

k=2

θi(τ , ρg ,k),

(5.12)ǫg ,2 ={

T r4096g

212∏12

k=1 cos(πρg ,k )∈ {−1, 1} when

∏12k=1 cos(πρg ,k) �= 0

0 when∏12

k=1 cos(πρg ,k) = 0

(5.13)ǫg ,3 = ǫg ,4 = 1,


SCA and see how the decomposition leads to mock modular forms relevant for the M22 moonshine which we will discuss in Sect. 8.

Recall (cf. [32]) that the N = 4 superconformal algebra contains subalgebras isomor-phic to the affine SU(2) and Virasoro Lie algebras. In a unitary representation the former of these acts with level m− 1, for some integer m > 1, and the latter with central charge c = 6(m− 1).

The unitary irreducible highest weight representations vN=4m;h,j are labeled by the

eigenvalues of L0 and 12 J30 acting on the highest weight state, which we denote by h

and j, respectively. Cf. [33, 34]. The superconformal algebra has two types of high-est weight Ramond sector representations: the massless (or BPS) representations with h = c

24 = m−14 and j ∈ {0, 12 , . . . ,

m−12 }, and the massive (or non-BPS) representations

with h > m−14 and j ∈ { 12 , 1, . . . ,

m−12 }. Their graded characters, defined as

are given by

and

in the massless and massive cases, respectively, [34]. In the above formulas, the function µm;j(τ , z) is defined by setting

and �1,1 is a meromorphic Jacobi form (cf. Sect. 8 of [19] for more on meromorphic Jacobi forms) of weight 1 and index 1 given by

Finally, we have used the theta functions

defined for all 2m ∈ Z>0 and r −m ∈ Z, and satisfying

Note that the vector-valued theta function θm = (θm,r), r −m ∈ Z/2mZ, is a vector-val-ued Jacobi form of weight 1/2 and index m satisfying

(6.1)chN=4m;h,j (τ , z) = trvN=4

m;h,j

((−1)J

30 yJ

30 qL0−c/24

),

(6.2)chN=4m;h,j (τ , z) = (�1,1(τ , z))

−1µm;j(τ , z)

(6.3)chN=4m;h,j (τ , z) = (�1,1(τ , z))

−1 qh−c24−

j2

m(θm,2j(τ , z)− θm,−2j(τ , z)

)

(6.4)µm;j(τ , z) = (−1)1+2j∑

k∈Zqmk2y2mk (yq

k)−2j + (yqk)−2j+1 + · · · + (yqk)1+2j

1− yqk,

(6.5)�1,1(τ , z) = −iθ1(τ , 2z) η(τ )

3

(θ1(τ , z))2= y+ 1

y− 1− (y2 − y−2)q + · · · .

(6.6)θm,r(τ , z) =∑

k=r (mod 2m)

e( k2 ) qk2/4myk ,

θm,r(τ , z) = θm,r+2m(τ , z) = e(m) θm,−r(τ ,−z).


where the Sθ and Tθ matrices are 2m× 2m matrices with entries

We will take m ∈ Z for the rest of this section. When we consider N = 2 decomposi-tions in the next section, we will use the theta function with half-integral indices.

From the above discussion, it is clear that the graded partition function of a module for the c = 6(m− 1) N = 4 SCA admits the following decomposition

Furthermore, from the identity

we arrive at

where

The rest of the components of F (m) = (F(m)r ), r ∈ Z/2mZ, are defined by setting

(6.7)

θm(τ , z) =√

1

2m

√i

τe(−m

τz2)Sθ .θm(− 1

τ, zτ)

= Tθ .θm(τ + 1, z)

= θm(τ , z + 1) = e(m(τ + 2z + 1))θm(τ , z + τ ),

(6.8)(Sθ )r,r′ = e( rr′

2m )e(−r+r′2 ), (Tθ )r,r′ = e(− r2

4m ) δr,r′ .

(6.9)ZN=4,m =

∑

n≥0,0≤r≤m−1r �=0 when n>0

c′r(n− r2

4m ) chN=4m;m−1

4 +n, r2(τ , z).

µm; r2 = (−1)r(r + 1)µm;0 + (−1)n−1r∑

n=1

n q−(r−n+1)2

4m (θm,r−n+1 − θm,−(r−n+1))

(6.10)ZN=4,m = (�1,1(τ , z))−1

c0 µm;0(τ , z)+

�

r∈Z/2mZ

F (m)r (τ ) θm,r(τ , z)

,

(6.11)F (m)r (τ ) =

∞∑

n=0

cr

(n− r2

4m

)qn−

r2

4m , 1 ≤ r ≤ m− 1,

(6.12)c0 =m−1∑

r=0

(−1)r (r + 1) c′r(− r2

4m

),

(6.13)cr

�n− r2

4m

�=

�m−1r′=r (−1)r

′−r(r′ + 1− r) c′r′�− r′2

4m

�, n = 0

c′r�n− r2

4m

�, n > 0.

(6.14)F (m)r (τ ) = −F

(m)−r (τ ) = F

(m)r+2m(τ ).


Recall that µm;0(τ , z) = −f(m)0 (τ , z)+ f

(m)0 (τ ,−z), a specialisation of the Appell–Lerch

sum

studied in [64], has the following relation to the modular group SL2(Z): let the (non-holomorphic) completion of µm;0(τ , z) be

Then µm;0 transforms like a Jacobi form of weight 1 and index m for SL2(Z)⋉ Z2. Here Sm = (Sm,r) is the vector-valued cusp form for SL2(Z) whose components are given by the unary theta functions

For later use, note that the theta series Sm,r(τ ) is defined for all 2m ∈ Z and r −m ∈ Z/2mZ.

The way in which the functions Z(m) and µm;0 transform under the Jacobi group shows that the non-holomorphic function

∑r∈Z/2mZ

F(m)r (τ ) θm,r(τ , z) transforms as a Jacobi

form of weight 1 and index m under SL2(Z)⋉ Z2, where

In other words, F (m) = (F(m)r ), r ∈ Z/2mZ is a vector-valued mock modular form with a

vector-valued shadow c0 Sm, whose r-th component is given by Sm,r(τ ), with the multi-plier for SL2(Z) given by the inverse of the multiplier system of Sm (cf. (6.8)).

Now we are ready to apply the above discussion to the U(1)-graded Ramond sector partition function of the theory, discussed in Sect. 5. Recall that in this case we have c = 12, so m = 3 in (6.2) and (6.3). The N = 4 decomposition of (5.1) gives

(6.15)f (m)u (τ , z) =

∑

k∈Z

qmk2y2mk

1− yqke(−u)

(6.16)

µm;0(τ , τ , z) = µm;0(τ , z)− e

(− 1

8

)1√2m

×∑

r∈Z/2mZ

θm,r(τ , z)

∫i∞

−τ

(τ ′ + τ )−1/2Sm,r(−τ ′) dτ ′ .

Sm,r(τ ) =∑

k=r (mod 2m)

e(k2

)k qk

2/4m = 1

2π i

∂

∂zθm,r(τ , z)|z=0.

F (m)r (τ ) = F (m)

r (τ )+ c0e(− 1

8

) 1√2m

∫ i∞

−τ

(τ ′ + τ )−1/2Sm,r(−τ ′) dτ ′.

(6.17)

Z(τ , z) = 21 chN=4

3; 12,0+ ch

N=4

3; 12,1+

(560 ch

N=4

3; 32, 12

+ 8470 chN=4

3; 52, 12

+ 70576 chN=4

3; 72, 12

+ · · ·

)

+

(210 ch

N=4

3; 32,1+ 4444 ch

N=4

3; 52,1+ 42560 ch

N=4

3; 72,1+ · · ·

)

(6.18)= (�1,1(τ , z))−1

24µ3;0(τ , z)+

�

r∈Z/6Zhr(τ )θ3,r(τ , z)


where . . . stand for terms with expansion �−11,1 q

αyβ with α − β2/12 > 3. More Fou-rier coefficients of the functions hr(τ ) are recorded in “Appendix C”, where h = hg for [g] = 1A. Note that all the graded multiplicities c′r(n− r2

12 ) appear to be non-negative. Of course, this is guaranteed by the fact that V is a module for the N = 4 SCA as shown in Sect. 3. In particular, the Fourier coefficients of hr(τ ) appear to be all non-negative apart from that of the polar term −2q−1/12 in h1.

From the above discussion we see that h = (hr), for r ∈ Z/6Z, is a weight 1/2 vector-valued mock modular form for SL2(Z) with 6 components (but just 2 linearly independ-ent components, since h0 = h3 = 0, h−1 = −h1, and h−2 = −h2), with shadow given by 24 S3, and multiplier system inverse to that of S3.

This is to be contrasted with the elliptic genus of a generic non-chiral super conformal field theory. For example, the sigma model of a K3 surface has c = 6, and the elliptic genus is given by

where . . . stand for terms with expansion �−11,1 q

αyβ with α − β2/8 > 3. In this case, the coefficient multiplying the massless character chN=4

2; 14 ,12 is negative, arising from the Witten

index of the right-moving massless multiplets paired with the representation vN=42; 14 ,

12

of the left-moving N = 4 SCA.

In Sect. 3 we have shown that the theory under consideration, as a module for the N = 4 SCA, admits a faithful action via automorphisms by a group G, as long as G is a subgroup of Co0 fixing at least a 3-plane. For any such g ∈ G, the g-twined graded parti-tion function Zg (τ , z) is given by (5.10), and from the fact that the action of g commutes with the N = 4 SCA, we expect Zg (τ , z) to admit a decomposition

Moreover, the coefficients of

must be characters of the G-module

(6.19)

EG(τ , z;K3) = TrHRR(−1)FL+FRy

J0qL0−c/24

qL0−c/24

= 20 chN=4

2; 14,0− 2 ch

N=4

2; 14, 12

+(90 ch

N=4

2; 54, 12

+ 462 chN=4

2; 94, 12

+ 1540 chN=4

2; 134, 12

+ · · ·)

(6.20)= (�1,1(τ , z))

−1{24µ2;0(τ , z)+ (θ2,1(τ , z)− θ2,−1(τ , z))

× (−2q−1/8 + 90q7/8 + 462q15/8 + 1540q23/8 + · · · )},

(6.21)Zg (τ , z) = (�1,1(τ , z))−1

(Tr24g) µ3;0(τ , z)+

�

r∈Z/6Zhg ,r(τ )θ3,r(τ , z)

.

(6.22)hg ,r(τ ) = arq−r2/12 +

∞∑

n=1

(TrVGr,ng) qn−r2/12

(6.23)VG =⊕

r=1,2

∞⊕

n=1

VGr,n


arising from the orbifold theory discussed in Sect. 2.Indeed, the multiplicities of the N = 4 multiplets in the decomposition (6.17) are sug-

gestive of the following group theoretic interpretationb: the 21 h = 1/2, j = 0 massless representations transform as the 21-dimensional irreducible representation of M22, and similarly, the 560 h = 3/2, j = 1/2 massive representations transform as χ10 + χ11 (see “Appendix B”), or “280+ 280”, under M22, etc.

We have explicitly computed the first 30 or so coefficients of each q-series hg ,r(τ ) for all conjugacy classes [g] of G, for G = M22 and G = U4(3). These can be found in the tables in “Appendix C”. Subsequently, we compute the first 30 or so G-modules VG

r,n in terms of their decompositions into irreducible representations. They can be found in the tables in “Appendix D”.

Finally we would like to discuss the mock modular property of the functions hg = (hg ,r). Recall that the Hecke congruence subgroups of SL2(Z) are defined as

We expect Zg to be a weak Jacobi form of weight zero and index 2 (possibly with multi-plier) for the group Ŵ0(og )⋉ Z2, where og is the order of the group element g ∈ G. This can be verified explicitly from the expression (5.10). Repeating the similar arguments as above, we conclude that each vector-valued function hg is a vector-valued mock modular form of weight 1/2 with shadow (Tr24g)S3 for the congruence subgroup Ŵ0(og ). Note that (Tr24g) �= 0 for all g ∈ M22 which are the cases of our main interest. For these cases the multiplier of hg is again given by the inverse of the multiplier system of S3, now restricted to Ŵ0(og ).

In this section we have analyzed the decomposition of the Ramond sector of our orbi-fold theory into irreducible modules for the N = 4 SCA, and we have demonstrated that the generating functions of irreducible N = 4 SCA module multiplicities furnish a vector-valued mock modular form. We have also demonstrated that these multiplici-ties are dimensions of modules for subgroups G < Co0 that point-wise fix a 3-plane in 24, and we have analyzed the modularity of the resulting, g-twined multiplicity generat-ing functions, for g ∈ G. We have verified that each such g-twining results in a vector-valued mock modular form with a specified shadow function. In the next section we will present directly analogous considerations for N = 2 superconformal structures arising from 2-planes in 24.

7 The N = 2 decompositionsAs discussed in Sect. 4, the theory presented in Sect. 2 can be regarded as a module for an N = 2 SCA as well as for an N = 4 SCA. Moreover, for every subgroup G < Co0 fixing a 2-plane there is an N = 2 SCA commuting with the action of G on the theory. As a result, and as we will now demonstrate, the decomposition of the partition func-tion (5.10) twined by elements of G into N = 2 characters leads to sets of vector-valued mock modular forms, now of weight 1 / 2 and index 3 / 2, which are the graded charac-ters of an infinite-dimensional G-module inherited from the Co0-module structure on V (cf. Sect. 5).

(6.24)Ŵ0(N ) ={(

a bc d

)∈ SL2(Z) | c=0 mod N

}.


To see what these (vector-valued) mock modular forms hg = (hg ,j) are, let us start by recalling the characters of the irreducible representations of the N = 2 SCA. For the SCA with central charge c = 3(2ℓ+ 1) = 3c, the unitary irreducible high-est weight representations vN=2

ℓ;h,Q are labeled by the two quantum numbers h and Q which are the eigenvalues of L0 and J0, respectively, when acting on the highest weight state [22, 51]. Just as in the N = 4 case, there are two types of Ramond sector high-est weight representations: the massless (or BPS) representations with h = c

24 = c8 and

Q ∈ {− c2 + 1,− c

2 + 2, . . . , c2 − 1, c2 }, and the massive (or non-BPS) representations with h > c

8 and Q ∈ {− c2 + 1,− c

2 + 2, . . . , c2 − 2, c2 − 1, c2 }, Q �= 0. From now on we will con-centrate on the case when ℓ is half-integral, and hence c and c are even.

The graded characters, defined as

are given by

for the massive representations, and

for the massless representations (with Q �= c2). The character chN=2

ℓ;c/24,Q(τ , z) for Q = c2

is given in (7.5). In the above formula, we have used the Appell–Lerch sum (6.15) and defined

Note that the above characters transform according to the rule

under charge conjugation.From the relation between the massless and massive characters

(7.1)chN=2ℓ;h,Q(τ , z) = trvN=2

ℓ;h,Q

((−1)J

30 yJ

30 qL0−c/24

),

(7.2)chN=2

ℓ;h,Q(τ , z) = e( ℓ2 )(�1,− 12(τ , z))−1qh−

c24−

j2

4ℓ θℓ,j(τ , z),

j = sgn(Q) (|Q| − 1/2),

(7.3)ch

N=2ℓ;c/24,Q(τ , z) = e

(ℓ+Q+1/2

2

)(�

1,− 12

(τ , z))−1 yQ+ 12 f (ℓ)u (τ , z + u),

u = 12+ (1+2Q)τ

4ℓ,

�1,− 12= −i

η(τ)3

θ1(τ , z)= 1

y1/2 − y−1/2+ q (y1/2 − y−1/2)+O(q2).

chN=2ℓ;c/24,Q(τ , z) = chN=2

ℓ;c/24,−Q(τ ,−z)

(7.4)

chN=2ℓ;c/24,Q + chN=2

ℓ;c/24,−Q = q−n|Q|−1∑

k=0

(−1)k(chN=2

ℓ,n+c/24,Q−k + chN=2ℓ,n+c/24,k−Q

)

+ 2(−1)QchN=2ℓ;c/24,0, n > 0, |Q| ≤ ℓ,

(7.5)

chN=2

ℓ;c/24, c2= q−n

chN=2

ℓ;n+c/24, c2+

c2−1�

k=1

(−1)k�chN=2

ℓ,n+c/24, c2−k+ chN=2

ℓ,n+c/24,k− c2

�

+ (−1)c2 chN=2

ℓ;c/24,0,


as well as the charge conjugation symmetry of the theory, we expect the U(1)-graded Ramond sector partition function of a theory that is invariant under charge conjugation to admit a decomposition

when the N = 2 SCA has even central charge, c = 3(2ℓ+ 1). In the last equation, we have defined

and

Similar to the case of massless N = 4 characters, through its relation to the Appell–Lerch sum, µℓ;0 admits a completion which transforms as a weight one, half-integral index Jac-obi form under the Jacobi group. More precisely, define µℓ;0 by replacing µm;0 with µℓ;0 and the integer m with the half-integral ℓ in (6.16). Then µℓ;0 transforms like a Jacobi form of weight 1 and index ℓ under the group SL2(Z)⋉ Z2. Following the same computation as in the previous section, we hence conclude that F (ℓ) = (F

(ℓ)j ), where j − 1/2 ∈ Z/2ℓZ, is

a vector-valued mock modular form with a vector-valued shadow C0 Sℓ = C0(Sℓ,j(τ )).Now we are ready to apply the above discussion to the U(1)-graded R sector partition

function of the orbifold theory of Sect. 2. The N = 2 decomposition gives

(7.6)

ZN=2,ℓ = C

′0 ch

N=2ℓ; c

24,0(τ , z)+

∑

n≥0

C′ℓ

(n− ℓ

4

)ch

N=2

ℓ; c

24+n,ℓ+ 1

2

(τ , z) < error−

+∑

n≥0,j∈{ 12, 32,...,ℓ−1}

C′j

(n− j

2

4ℓ

)(ch

N=2

ℓ; c

24+n,j+ 1

2

(τ , z)+ chN=2

ℓ; c

24+n,−

(j+ 1

2

)(τ , z)

)

(7.7)= e�ℓ2

�(�1,− 1

2)−1

C0 µℓ;0(τ , z)+

�

j−ℓ∈Z/2ℓZF(ℓ)j (τ )θℓ,j(τ , z)

µℓ;0 = e( 14 ) y1/2f (ℓ)u (τ ,u+ z), u = 1

2+ τ

4ℓ,

(7.8)F(ℓ)j (τ ) = F

(ℓ)−j (τ ) = F

(ℓ)j+2ℓ(τ ) =

∑

n≥0

Cj

(n− j2

4ℓ

)qn−

j2

4ℓ ,

(7.9)C0 = C ′0 + 2

∑

j∈{ 12 ,32 ,...,ℓ}

(−1)j+12C ′

j

(− j2

4ℓ

),

(7.10)Cj

�n− j2

4ℓ

�=

�ℓ−jk=0(−1)kC ′

j+k

�− (j+k)2

4ℓ

�n = 0

C ′j

�n− j2

4ℓ

�n > 0.

(7.11)

Z(τ , z) = 23 chN=232; 12,0+ ch

N=232; 12,2+

(770

(ch

N=232; 32,1+ ch

N=232; 32,−1

)

+13915

(ch

N=232; 52,1+ ch

N=232; 52,−1

)+ · · ·

)

+(231 ch

N=232; 32,2+ 5796 ch

N=232; 52,2+ . . .

)


where . . . denote the terms with expansion �−1

1,− 12

qαyβ with α − β2/6 > 2. Again, we observe that all the multiplicities of the representations with characters chN=2

32 ;h,Q

appear to be non-negative, consistent with our construction of V as an N = 2 SCA module.

In general, from the previous sections we have seen that the graded partition func-tion twined by any element g of a subgroup G of Co0 should admit a decomposition into N = 2 characters. We write

Moreover, from the discussion in Sect. 2 we have seen that

and the coefficients of these functions

are given by characters of a G-module

which descends from the orbifold theory in Sect. 2. In particular, for any n, the G-mod-ule V G

−1/2,n is the dual of V G1/2,n. From the above discussion, we conclude that hg = (hg ,j)

is a vector-valued mock modular form for Ŵ0(og ) with shadow (Tr24g)S3/2. Recall that (Tr24g) �= 0 for all g ∈ M22 which are the cases of our main interest. For these case the multiplier of hg is given by the inverse of the multiplier system of S3/2, restricted to Ŵ0(og ).

To summarize, we have analyzed the decomposition of the Ramond sector of our orbi-fold theory into irreducible modules for the N = 2 SCA in this section, and we have demonstrated that the generating functions of irreducible N = 2 SCA module multi-plicities also furnish a vector-valued mock modular form. We have demonstrated that these multiplicities are dimensions of modules for subgroups G < Co0 that point-wise fix a 2-plane in 24, and we have observed that the resulting, g-twined multiplicity gen-erating functions, for g ∈ G, are vector-valued mock modular forms with a certain extra symmetry, relating the components labelled by ±1/2 by complex conjugation.

8 Mathieu moonshineIn the previous sections we have seen that the orbifold theory described in Sect. 2 leads to infinite-dimensional G-modules underlying a set of vector-valued mock modular

(7.12)

= e(34

)�−1

1,− 12

(24 µ 3

2 ;0+

(−q−

124 + 770 q

2324 + 13915 q

4724 + c . . .

)(θ 32 ,

12+ θ 3

2 ,−12

)

+ (q−38 + 231 q

58 + 5796q

138 + · · · ) θ 3

2 ,32

)

(7.13)Zg (τ , z) = e( 34 )�−1

1,− 12

(Tr24g) µ 3

2 ;0(τ , z)+

�

j∈{1/2,−1/2,3/2}hg ,j(τ )θ 3

2 ,j

.

(7.14)hg ,1/2(τ ) = hg ,−1/2(−τ ),

(7.15)hg ,j(τ ) = ajq−j2/6 +

∞∑

n=1

(TrV Gr,ng) qn−j2/6

(7.16)V G =⊕

j=−1/2,1/2,3/2

∞⊕

n=1

V Gj,n


forms from its N = 4 (N = 2) structures for any subgroup G of Co0 fixing at least a 3-plane (2-plane) in 24. In this section we will discuss a natural property of the vector-valued mock modular forms that distinguishes subgroups of M24 from other 3-plane (2-plane) fixing subgroups. These considerations lead to mock modular Mathieu moon-shine involving distinguished vector-valued mock modular forms of weight 1 / 2.

Recall the celebrated genus zero condition of monstrous moonshine, which states that the monstrous McKay–Thompson series are Hauptmoduls with only a polar term q−1 at the cusp represented by i∞, and no poles at any other cusps. To be more pre-cise, denote by Tg (τ ) =

∑n≥−1 q

ntrV

♮ng the graded character of the moonshine module

V ♮ =⊕

n≥−1 V♮n of Frenkel–Lepowsky–Meurman [37]. Then Tg (τ ) is a function invari-

ant under the action of a particular Ŵg < SL2(R) (specified in [15]), such that

Similarly, in Mathieu moonshine [35], it follows from the results of [7] that if g ∈ M24 and Zg denotes the g-twined K3 elliptic genus then

(cf. (5.4)) for some cg , cg ,γ ∈ C. In other words, the τ → i∞ limit of Zg |0,1γ is a z-inde-pendent constant whenever γ is not in the invariance group Ŵg. (Note that Ŵg is always a subgroup of SL2(Z) for g ∈ M24).

A natural question is therefore: among the subgroups of Co0 fixing 2- or 3-planes for which we have constructed a module in this work, for which of these do the associated modular objects satisfy a condition analogous to the preceding cases of moonshine described above? We will see presently that the Mathieu groups M23, M22, M12 and M11 are distinguished in our setting, in that the graded characters of their respective modules yield weight zero weak Jacobi forms satisfying the conditions

for some cg , cg ,γ ∈ C. On the other hand, all the other groups mentioned in Sect. 4 con-tain elements g for which the condition (ii) in (8.3) is not satisfied. Thus the conditions (8.3) single out the Mathieu groups as the sporadic simple subgroups of Co0 with this moonshine property. The constructions we have presented in this paper provide con-crete realizations of the underlying mock modular Mathieu moonshine modules.

Note that the conditions (8.3) impose restrictions on the degrees of the poles (if any) of the mock modular forms hg, hg (cf. Sects. 6, 7) at all cusps. In fact, for the case that G is a copy of M22 or M11 preserving N = 4 supersymmetry, the corresponding mock modular forms hg only have poles at the infinite cusp. This property also holds for the mock modular forms attached to M24 via N = 4 decomposition of the twined K3 ellip-tic genera of Mathieu moonshine, satisfying (8.2), as was demonstrated in [7]. In more

(8.1)(i) qTg (τ ) = O(1) as τ → i∞, and

(ii) Tg (γ τ) = O(1) as τ → i∞whenever γ ∈ SL2(Z) and γ∞ �∈ Ŵg∞.

(8.2)(i) Zg (τ , z) = cg + y+ y−1 as τ → i∞, and

(ii) Zg |0,1γ (τ , z) = cg ,γ as τ → i∞ whenever γ ∈ SL2(Z) and γ∞ �∈ Ŵg∞,

(8.3)(i) Zg (τ , z) = cg + y2 + y−2 as τ → i∞, and

(ii) Zg |0,2γ (τ , z) = cg ,γ as τ → i∞ whenever γ ∈ SL2(Z) and γ∞ �∈ Ŵg∞,


physical terms, (8.2) and (8.3) can be interpreted as the condition that the elliptic genus of any cyclic orbifold of the theory receives no contributions from U(1)-charged ground states in twisted sectors.

To investigate the behaviour of Zg at cusps other than i∞, we first note that for any positve integer N,

where

and ǫi,N = ǫi when 2|N , and

otherwise. The above expressions can be derived from the transformation laws of Jacobi theta functions under SL2(Z).

Near the infinite cusp, τ → i∞, the different contributions have the following leading behaviour:

with

Zg |0,2( 1 0N 1

)(τ , z) = e

(N

12∑

ℓ=1

ρ2ℓ

2

)4∑

i=1

ǫi,N (−1)i+1�i,N (τ , z),

(8.4)�i,N (τ , z) =θi(τ , 2z)

η12(τ )

12∏

k=2

e

(ρ2kN

2

2τ

)θi(τ , ρk + Nρkτ ),

(8.5)

ǫ1,N = −ǫ1ǫ2,N = −ǫ3 = −1

ǫ3,N = −ǫ2ǫ4,N = −ǫ4 = −1

(8.6)

�1,N (τ , z) = e

�12+

12�

k=1

( 12− ρk)⌊Nρk⌋ − ρk

2

�qfN ,1(�g )/2(y− y−1) [1+O(q1/og )]

�2,N (τ , z) = e

�12�

k=1

−ρk⌊Nρk⌋ − ρk2

�qfN ,1(�g )/2(y+ y−1) [1+ O(q1/og )]

�3,N (τ , z) = e

�−

12�

k=1

ρk⌊ 12 + Nρk⌋�qfN ,2(�g )/2

�1+ q1/2(y2 + y−2)

�

×

12�

k=2

�1+ e(−ρk) q

⌊ 12+Nρk⌋+

12−Nρk

� ⌊ 12+Nρk⌋�

n=1

�1+ e(ρk)q

12+Nρk−n

�

× [1+ O(q1/og )]

�4,N (τ , z) = e

�−

12�

k=1

( 12+ ρk)⌊ 12 + Nρk⌋

�qfN ,2(�g )/2

�1− q1/2(y2 + y−2)

�

×

12�

k=2

�1− e(−ρk) q

⌊ 12+Nρk⌋+

12−Nρk

� ⌊ 12+Nρk⌋�

n=1

�1− e(ρk) q

12+Nρk−n

�

× [1+ O(q1/og )]

(8.7)fN ,1(�g ) = −1+12∑

k=1

(Nρk − 1/2)2 + ⌊Nρk⌋(1+ ⌊Nρk⌋ − 2Nρk) ,


where ⌊x⌋ denotes the largest integer that is not greater than x.Comparing with the second condition in (8.3), we see that it is satisfied for γ =

( 1 0N 1

)

if and only if the �g satisfies

together with

for the case ǫ3,N ǫ4,N ǫ( 12∑12

k=1⌊ 12Nρk⌋) = 1, and

for the case ǫ3,N ǫ4,N ǫ( 12∑12

k=1⌊ 12Nρk⌋) = −1.Now we are left to check explicitly the condition (ii) in (8.3) for the various 2- and 3-plane

preserving subgroups discussed in Sect. 4. First, recall that, for n a positive integer, each cusp of Ŵ0(n) is represented by a rational number of the form u / v where u and v are coprime positive integers, v a divisor of n, and u / v is equivalent u′/v′ if and only if v = v′, and u = u′ mod (v, n/v). (note that the infinite cusp is also represented by 1 / n.) Via direct computation using the data of the eigenvalues of g ∈ Co0 acting on 24, we note that among all the groups we have considered in Sect. 4, the groups U4(3), U6(2) and McL all contain a conjugacy classe with Frame shape �g = 39/13, and HS has a conjugacy class with Frame shape �g = 55/11. One can explicitly check that f1,1(�g ) = 0 for these classes and hence the corresponding twining Zg |0,2

(1 01 1

)(τ , z) has a non-vanishing coefficient for q0y1 as

τ → i∞. (Note that the cusp at τ = 1 is not equivalent to the cusp at i∞ in these cases.) This excludes the groups U4(3), U6(2), McL and HS as candidates for moonshine satisfying (8.3).

For the subgroups of M24, a simple analysis of the cusp representatives of Ŵg = Ŵ(og ) shows that it is sufficient to verify (8.3) for γ =

( 1 0N 1

) for all N|n and N < n. For these

cases, we explicitly verify that (8.9)–(8.11) are satisfied and hence the moonshine condi-tion (8.3) is met.

We therefore conclude that we have established mock modular moonshine for all but the largest of the sporadic simple Mathieu groups, together with explicit constructions of the corresponding modules. Moreoever, the corresponding twined graded characters are mock modular forms arising from Jacobi forms satisfying the distinguishing condi-tions (8.3), which we may recognise as furnishing a natural analogue of the powerful principal modulus property (a.k.a. genus zero property) of monstrous moonshine.

The reader will note that many of the numbers which occur as dimensions of irreduc-ible representations of M23 also occur as dimensions of irreducible representations for M24. Indeed, looking at the Tables in “Appendix C”, one is tempted to guess that there is an alternative construction, or hidden symmetry in our model, which yields an M24

-module with the same graded dimensions. In fact, the procedure we have explained for computing twinings can be carried out for any element of M24, regarded as a subgroup

(8.8)fN ,2(�g ) = −1+12∑

k=1

N 2ρ2k − ⌊ 12 + Nρk⌋(2Nρk − ⌊ 12 + Nρk⌋) ,

(8.9)fN ,1(�g ) > 0,

(8.10)fN ,2(�g ) > −1,fN ,2(�g )

2+

∣∣∣∣1

2+ ⌊Nρk⌋ − Nρk

∣∣∣∣ ≥ 0

(8.11)fN ,2(�g ) ≥ 0


of Co0, for any such element fixes a 2-space in 24. However, there is no 2-space that is fixed by every element of a given copy of M24, and explicit computations reveal that any M24-module structure on the module we have constructed would have to involve virtual representations. This indicates that there is no direct extension to M24 of the Mathieu moonshine modules we have considered here, despite the prominent role M24 plays in incorporating the different groups of moonshine in the current setting. Nevertheless, there is a certain modification of our method for which M24 is now known to play a lead-ing role. We refer the reader to the next, and final section for a description of this.

9 DiscussionIn this paper we have demonstrated that, starting with the free field Co0 module of [24], one can construct explicit examples of modules for various subgroups G ⊂ Co0 which underlie certain mock modular forms. In particular, subgroups which preserve a 3-plane (respectively 2-plane) in the 24 give rise to N = 4 (N = 2) modules with G symmetry. This gives completely explicit examples of mock modular moonshine for the smaller Mathieu groups, where the modules are known, and where the twining functions are distinguished in a manner directly similar to Mathieu moonshine. Other examples, including modules for the sporadic groups McL and HS, are also described.

There are several future directions. We considered here the N = 2 and N = 4 extended chiral algebras, and the subgroups of Co0 that they preserve. Other extended chiral algebras may also yield interesting results. For instance, supersymmetric sigma models with target a Spin(7) manifold give rise to an extended chiral algebra [58], whose representations were studied in [45]. It is an extension of the N = 1 superconformal algebra where instead of adding a U(1) current (which extends the theory to an N = 2 superconformal theory), one chooses an additional Ising factor. Conjectural characters for the unitary, irreducible representations of this algebra were worked out in [2], where it was shown that there is a suggestive relation between the decomposition of the ellip-tic genus of a Spin(7) manifold into these characters and irreducible representations of finite groups. This connection was made precise in [12], where it was shown that the same c = 12 theory can be viewed as an SCFT with extended N = 1 symmetry, and thus yields theories with global symmetry groups M24, Co2, and Co3. The partition function twined by these symmetries, when decomposed into characters of the Spin(7) algebra, gives rise to two-component vector-valued mock modular forms encoding infinite-dimensional modules for the corresponding sporadic groups.

The motivation that led, eventually, to the present study was actually to find connec-tions between geometrical target manifolds associated to c = 12 conformal field theo-ries, and sporadic groups. The elliptic genera of Calabi-Yau fourfolds were computed in [54], for instance; their structure is reminiscent of some of the modules we have seen here, and we intend to further explore and describe some of these connections in a future publication. Likewise, hyperkähler fourfolds, as well as the Spin(7) manifolds mentioned above, provide a wide class of geometries where an analogue of the connec-tions between M24 and K3 may be sought.

Last but not least, there are suggestive connections between the trace functions in moonshine modules, and certain special properties of the underlying conformal field theory. Both the CFT appearing in monstrous moonshine and the Co0 module that


played a starring role in this paper appear to play special roles also in AdS3 quantum gravity, where they are candidates for CFT duals to pure (super)gravity [62]. The genus zero property of the twining functions in monstrous moonshine can be reformulated as a condition that these class functions should be expressed as Rademacher sums based on a fixed polar part [7, 29]; this latter description then applies uniformly to monstrous moonshine and Mathieu moonshine.

In this paper we demonstrate that a similar criterion also applies to our mock modu-lar Mathieu moonshines. In particular, we have shown that the Jacobi forms relevant for Mathieu moonshine display a specific asymptotic behaviour near non-infinite cusps, which, in physical terms, can be interpreted as a condition on orbifolds of the theory. Pursuing a deeper understanding of this property constitutes an enticing direction for the future.

10 EndnoteaSee however the recent work [26] which constructs the super vertex operator algebra

underlying the X = E38 case of umbral moonshine.

bThe observation that the decomposition into N = 4 characters of (a multiple of ) the function Z(τ , z) returns positive integers that are suggestive of representations of the Mathieu group M22 was first communicated privately by Jeff Harvey to J.D. in 2010.

Authors’ contributionsEach author contributing equally to all aspects of the production of this article. All authors read and approved the final manuscript.

Author details1 Institute of Physics and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Neth-erlands. 2 Department of Physics and SLAC, Stanford University, Stanford, CA 94305, USA. 3 Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA.

AcknowledgmentsWe are grateful to Jeff Harvey, Sander Mack-Crane and Daniel Whalen for conversations about related subjects. We are grateful to Daniel Baumann, and the anonymous referees, for very helpful comments on earlier drafts. We note that the appearance of dimensions of M22 representations in an N = 4 decomposition of the partition function of the model studied in this paper was described in correspondence from J. Harvey to J. Duncan in 2010. The expression (5.2) for Z(τ , z) first came to our attention during the course of conversations with S. Mack-Crane. We thank the Simons Center for Geometry and Physics, and in particular the organizers of the workshop on “Mock Modular Forms, Moonshine, and String Theory,” for hospitality when this work was initiated. S.K. is grateful to the Aspen Center for Physics for providing the rockies during the completion of this work. X.D., S.H. and S.K. are supported by the U.S. National Science Foundation Grant PHY-0756174, the Department of Energy under contract DE-AC02-76SF00515, and the John Templeton Founda-tion. J.D. is supported in part by the Simons Foundation (#316779) and by the U.S. National Science Foundation (DMS 1203162). T.W. is supported by a Research Fellowship (Grant number WR 166/1-1) of the German Research Foundation (DFG).

Miranda C N Cheng is on leave from CNRS, Paris.

Compliance with ethical guidelines

Competing interestsThe authors declare that they have no competing interests.

Appendix A: Jacobi theta functionsWe define the Jacobi theta functions θi(τ , z) as follows for q = e(τ ) and y = e(z):

(9.1)θ1(τ , z) = −iq1/8y1/2∞∏

n=1

(1− qn)(1− yqn)(1− y−1qn−1),

(9.2)

θ2(τ , z) = q1/8y1/2∞∏

n=1

(1− qn)(1+ yqn)(1+ y−1qn−1),


They transform in the following way under the group SL2(Z)⋉ Z2.

Here α(τ , z) =√−iτe( z

2

2τ ), and �,µ ∈ Z.The weight four Eisenstein series E4 can be written in terms of the Jacobi theta func-

tions as

(9.3)θ3(τ , z) =∞∏

n=1

(1− qn)(1+ y qn−1/2)(1+ y−1qn−1/2),

(9.4)θ4(τ , z) =∞∏

n=1

(1− qn)(1− y qn−1/2)(1− y−1qn−1/2).

(9.5)

θ1(τ , z) = i α−1(τ , z)θ1

(−1

τ,z

τ

)= e(−1/8) θ1(τ + 1, z)

= (−1)�+µe(12 (�

2τ + 2�z))θ1(τ , z + �τ + µ),

θ2(τ , z) = α−1(τ , z)θ4

(−1

τ,z

τ

)= e(−1/8) θ2(τ + 1, z)

= (−1)µe(12 (�

2τ + 2�z))θ2(τ , z + �τ + µ),

θ3(τ , z) = α−1(τ , z)θ3

(−1

τ,z

τ

)= θ4(τ + 1, z)

= e(12 (�

2τ + 2�z))θ3(τ , z + �τ + µ),

θ4(τ , z) = α−1(τ , z)θ2

(−1

τ,z

τ

)= θ3(τ + 1, z)

= (−1)�e(12 (�

2τ + 2�z))θ4(τ , z + �τ + µ).

(9.6)E4(τ ) =1

2

(θ2(τ , 0)

8 + θ3(τ , 0)8 + θ4(τ , 0)

8).

Table 1 Frame shapes and spinor characters for M22

[g] 1A 2A 3A 4A 4B 5A 6A 7A 7B 8A 11A 11B

�g 124

1828

1636 1

42444

142444

1454

12223362

1373

1373

122.4.8

21211

21211

2

Tr4096g 2,048 0 64 0 0 0 0 8 8 0 4 4

Table 2 Frame shapes and spinor characters for U4(3)

[g] 1A 2A 3A 3BCD 4AB 5A 6A 6BC 7AB 8A 9ABCD 12A

�g 124

1828 3

9

131636

142244 1

454 1

53.6

4

2412223362

1373

122.4.8

2 1393

321.2

23.12

2

42

Rg 2,048 0 −8 64 0 0 72 0 8 0 4 0

Appendix B: Character tablesB1: Frame shapes and spinor representations

See Tables 1, 2, 3, 4, 5, 6.


Table 3 Frame shapes and spinor characters for M23

[g] 1A 2A 3A 4A 5A 6A 7AB 8A 11AB 14AB 15AB 23AB

�g 124

1828

1636

142244 1

454

12223362

1373

122.4.8

21211

2 1.2.7.14 1.3.5.15 1.23

Tr4096g 2,048 0 64 0 0 0 8 0 4 0 4 2

Table 4 Frame shapes and spinor characters for McL

[g] 1A 2A 3A 3B 4A 5A 5B 6A 6B 7AB

�g 124

1828 3

9

131636

142244 5

5

11454 1

53.6

4

2412223362

1373

Tr4096g 2,048 0 −8 64 0 −4 0 72 0 8

[g] 8A 9AB 10A 11AB 12A 14AB 15AB 30AB

�g 122.4.8

2 1393

32135.10

2

221211

2 1.223.12

2

421.2.7.14 1

215

2

3.5

2.3.5.30

6.10

Tr4096g 0 4 20 4 0 0 2 2

Table 5 Frame shapes and spinor characters for HS

[g] 1A 2A 2B 3A 4A 4BC 5A 5BC 6A 6B

�g 124

1828

212

1636 2

644

14142244 5

5

11454

2363

12223362

Tr4096g 2,048 0 0 64 0 0 −4 0 0 0

[g] 7A 8ABC 10A 10B 11AB 12A 15A 20AB

�g 1373

122.4.8

2 135.10

2

222210

21211

2 124.6

212

321.3.5.15 1.2.10.20

4.5

Tr4096g 8 0 20 0 4 0 4 0

Table 6 Frame shapes and spinor characters for U6(2)

[g] 1A 2A 2B 2C 3A 3B 3C 4A 4B 4CDE 4F 4G

�g 124 2

16

181828

212

1636 3

9

131636 1

848

2848

24142244 2

444

142244

Tr4069g 2048 0 0 0 64 –8 64 256 0 0 0 0

[g] 5A 6AB 6C 6D 6E 6F 6G 6H 7A 8A 8BCD

�g 1454 1.6

6

22332464

1232153.6

4

24122232621

42.6

5

34122232622363

1373 1

484

2242122.4.8

2

Tr4096g 0 0 0 72 0 0 0 0 8 32 0

[g] 9ABC 10A 11AB 12AB 12C 12DE 12FGH 12I 15A 18AB

�g 1393

32122.10

3

521211

2 2.3312

3

1.4.6312324212

2

22621312

3

2.3.4.6

1.223.12

2

422.4.6.12 1.3.5.15 1.2.18

2

6.9

Tr4096g 4 0 4 4 16 12 0 0 4 0


B2: I

rred

ucib

le re

pres

enta

tions

See

Tabl

es 7

, 8, 9

, 10,

11,

12,

13,

14.

Tabl

e 7

Char

acte

r tab

le o

f M22. b

p=

(−1+

i√p)/2

[g]

1A2A

3A4A

4B5A

6A7A

7B8A

11A

11B

[g2]

1A1A

3A2A

2A5A

3A7A

7B4A

11B

11A

[g3]

1A2A

1A4A

4B5A

2A7B

7A8A

11A

11B

[g5]

1A2A

3A4A

4B1A

6A7B

7A8A

11A

11B

[g7]

1A2A

3A4A

4B5A

6A1A

1A8A

11B

11A

[g11]

1A2A

3A4A

4B5A

6A7A

7B8A

1A1A

χ1

11

11

11

11

11

11

χ2

215

31

11

−1

00

−1

−1

−1

χ3

45−

30

11

00

b7

b7

−1

11

χ4

45−

30

11

00

b7

b7

−1

11

χ5

557

13

−1

01

−1

−1

10

0

χ6

993

03

−1

−1

01

1−

10

0

χ7

154

101

−2

2−

11

00

00

0

χ8

210

23

−2

−2

0−

10

00

11

χ9

231

7−

3−

1−

11

10

0−

10

0

χ10

280

−8

10

00

10

00

b11

b11

χ11

280

−8

10

00

10

00

b11

b11

χ12

385

1−

21

10

−2

00

10

0


Tabl

e 8

Char

acte

r tab

le o

f U4(3). bp=

(−1+

i√p)/2

[g]

1A2A

3A3B

3C3D

4A4B

5A6A

6B6C

7A7B

8A9A

9B9C

9D12

A

[g2]

1A1A

3A3B

3C3D

2A2A

5A3A

3B3C

7A7B

4A9B

9A9D

9C6A

[g3]

1A2A

1A1A

1A1A

4A4B

5A2A

2A2A

7B7A

8A3A

3A3A

3A4A

[g5]

1A2A

3A3B

3C3D

4A4B

1A6A

6B6C

7B7A

8A9B

9A9D

9C12

A

[g7]

1A2A

3A3B

3C3D

4A4B

5A6A

6B6C

1A1A

8A9A

9B9C

9D12

A

χ1

11

11

11

11

11

11

11

11

11

11

χ2

215

−6

33

31

11

2−

1−

10

0−

10

00

0−

2

χ3

353

88

−1

−1

3−

10

00

30

0−

12

2−

1−

10

χ4

353

8−

18

−1

3−

10

03

00

0−

1−

1−

12

20

χ5

9010

99

90

−2

20

11

1−

1−

10

00

00

1

χ6

140

125

−4

−4

54

00

−3

00

00

0−

1−

1−

1−

11

χ7

189

−3

270

00

51

−1

30

00

01

00

00

−1

χ8

210

221

33

3−

2−

20

5−

1−

10

00

00

00

1

χ9

280

−8

1010

11

00

0−

2−

21

00

01+

3b3

1+

3b3

11

0

χ10

280

−8

1010

11

00

0−

2−

21

00

01+

3b3

1+

3b3

11

0

χ11

280

−8

101

101

00

0−

21

−2

00

01

11+

3b3

1+

3b3

0

χ12

280

−8

101

101

00

0−

21

−2

00

01

11+

3b3

1+

3b3

0

χ13

315

11−

918

−9

0−

1−

10

−1

2−

10

01

00

00

−1

χ14

315

11−

9−

918

0−

1−

10

−1

−1

20

01

00

00

−1

χ15

420

4−

396

6−

34

00

1−

2−

20

00

00

00

1

χ16

560

−16

−34

22

20

00

22

20

00

−1

−1

−1

−1

0

χ17

640

0−

8−

8−

81

00

00

00

b7

b7

01

11

10

χ18

640

0−

8−

8−

81

00

00

00

b7

b7

01

11

10

χ19

729

90

00

0−

31

−1

00

01

1−

10

00

00

χ20

896

032

−4

−4

−4

00

10

00

00

0−

1−

1−

1−

10


Tabl

e 9

Char

acte

r tab

le o

f M23. b

p=

(−1+

i√p)/2

[g]

1A2A

3A4A

5A6A

7A7B

8A11

A11

B14

A14

B15

A15

B23

A23

B

[g2]

1A1A

3A2A

5A3A

7A7B

4A11

B11

A7A

7B15

A15

B23

A23

B

[g3]

1A2A

1A4A

5A2A

7B7A

8A11

A11

B14

B14

A5A

5A23

A23

B

[g5]

1A2A

3A4A

1A6A

7B7A

8A11

A11

B14

B14

A3A

3A23

B23

A

[g7]

1A2A

3A4A

5A6A

1A1A

8A11

B11

A2A

2A15

B15

A23

B23

A

[g11]

1A2A

3A4A

5A6A

7A7B

8A1A

1A14

A14

B15

B15

A23

B23

A

[g23]

1A2A

3A4A

5A6A

7A7B

8A11

A11

B14

A14

B15

A15

B1A

1A

χ1

11

11

11

11

11

11

11

11

1

χ2

226

42

20

11

00

0−

1−

1−

1−

1−

1−

1

χ3

45−

30

10

0b7

b7

−1

11

−b7

−b7

00

−1

−1

χ4

45−

30

10

0b7

b7

−1

11

−b7

−b7

00

−1

−1

χ5

230

225

20

1−

1−

10

−1

−1

11

00

00

χ6

231

76

−1

1−

20

0−

10

00

01

11

1

χ7

231

7−

3−

11

10

0−

10

00

0b15

b15

11

χ8

231

7−

3−

11

10

0−

10

00

0b15

b15

11

χ9

253

131

1−

21

11

−1

00

−1

−1

11

00

χ10

770

−14

5−

20

10

00

00

00

00

b23

b23

χ11

770

−14

5−

20

10

00

00

00

00

b23

b23

χ12

896

0−

40

10

00

0b11

b11

00

11

−1

−1

χ13

896

0−

40

10

00

0b11

b11

00

11

−1

−1

χ14

990

−18

02

00

b7

b7

00

0b7

b7

00

11

χ15

990

−18

02

00

b7

b7

00

0b7

b7

00

11

χ16

1035

270

−1

00

−1

−1

11

1−

1−

10

00

0

χ17

2024

8−

10

−1

−1

11

00

01

1−

1−

10

0


Tabl

e 10

Cha

ract

er ta

ble

of M

cL. b

p=

(−1+

i√p)/2

, a=

1+

3b3

[g]

1A2A

3A3B

4A5A

5B6A

6B7A

7B8A

9A9B

10A

11A

11B

12A

14A

14B

15A

15B

30A

30B

[g2]

1A1A

3A3B

2A5A

5B3A

3B7A

7B4A

9B9A

5A11

B11

A6A

7A7B

15A

15B

15A

15B

[g3]

1A2A

1A1A

4A5A

5B2A

2A7B

7A8A

3A3A

10A

11A

11B

4A14

B14

A5A

5A10

A10

A

[g5]

1A2A

3A3B

4A1A

1A6A

6B7B

7A8A

9B9A

2A11

A11

B12

A14

B14

A3A

3A6A

6A

[g7]

1A2A

3A3B

4A5A

5B6A

6B1A

1A8A

9A9B

10A

11B

11A

12A

2A2A

15B

15A

30B

30A

[g11]

1A2A

3A3B

4A5A

5B6A

6B7A

7B8A

9B9A

10A

1A1A

12A

14A

14B

15B

15A

30B

30A

χ1

11

11

11

11

11

11

11

11

11

11

11

11

χ2

226

−5

42

−3

23

01

10

11

10

0−

1−

1−

10

0−

2−

2

χ3

231

715

6−

16

17

−2

00

−1

00

20

0−

10

00

02

2

χ4

252

289

94

22

11

00

00

0−

2−

1−

11

00

−1

−1

11

χ5

770

−14

−13

5−

2−

50

71

00

0−

1−

11

00

10

0b15

b15

b15

b15

χ6

770

−14

−13

5−

2−

50

71

00

0−

1−

11

00

10

0b15

b15

b15

b15

χ7

896

032

−4

0−

41

00

00

0−

1−

10

b11

b11

00

02

20

0

χ8

896

032

−4

0−

41

00

00

0−

1−

10

b11

b11

00

02

20

0

χ9

1750

70−

513

20

0−

51

00

0−

2−

20

11

−1

00

00

00

χ10

3520

64−

4410

0−

50

4−

2−

1−

10

11

−1

00

01

11

1−

1−

1

χ11

3520

−64

−44

100

−5

0−

42

−1

−1

01

11

00

0−

1−

11

11

1

χ12

4500

2045

−9

40

05

−1

−1

−1

00

00

11

1−

1−

10

00

0

χ13

4752

−48

540

02

2−

60

−1

−1

00

02

00

01

1−

1−

1−

1−

1

χ14

5103

630

03

3−

20

00

01

00

3−

1−

10

00

00

00

χ15

5544

−56

369

019

−1

41

00

00

0−

10

00

00

11

−1

−1

χ16

8019

−45

00

3−

6−

10

0−b7

−b7

−1

00

00

00

−b7

−b7

00

00

χ17

8019

−45

00

3−

6−

10

0−b7

−b7

−1

00

00

00

−b7

−b7

00

00

χ18

8250

1015

6−

20

0−

5−

2−b7

−b7

00

00

00

1b7

b7

00

00

χ19

8250

1015

6−

20

0−

5−

2−b7

−b7

00

00

00

1b7

b7

00

00

χ20

9625

105

40−

5−

30

00

30

0−

11

10

00

00

00

00

0


[g]

1A2A

3A3B

4A5A

5B6A

6B7A

7B8A

9A9B

10A

11A

11B

12A

14A

14B

15A

15B

30A

30B

χ21

9856

0−

80−

80

61

00

00

0a

a0

00

00

00

00

0

χ22

9856

0−

80−

80

61

00

00

0a

a0

00

00

00

00

0

χ23

1039

5−

2127

0−

1−

50

30

00

10

0−

10

0−

10

0b15

b15

−b15

−b15

χ24

1039

5−

2127

0−

1−

50

30

00

10

0−

10

0−

10

0b15

b15

−b15

−b15

Tabl

e 10

con

tinu

ed


Tabl

e 11

Cha

ract

er ta

ble

of HS.

bp=

(−1+

i√p)/2

, ap=

i√p

[g]

1A2A

2B3A

4A4B

4C5A

5B5C

6A6B

7A8A

8B8C

10A

10B

11A

11B

12A

15A

20A

20B

[g2]

1A1A

1A3A

2A2A

2A5A

5B5C

3A3A

7A4B

4C4C

5A5B

11B

11A

6B15

A10

A10

A

[g3]

1A2A

2B1A

4A4B

4C5A

5B5C

2B2A

7A8A

8B8C

10A

10B

11A

11B

4A5B

20A

20B

[g5]

1A2A

2B3A

4A4B

4C1A

1A1A

6A6B

7A8A

8B8C

2A2B

11A

11B

12A

3A4A

4A

[g7]

1A2A

2B3A

4A4B

4C5A

5B5C

6A6B

1A8A

8B8C

10A

10B

11B

11A

12A

15A

20A

20B

[g11]

1A2A

2B3A

4A4B

4C5A

5B5C

6A6B

7A8A

8B8C

10A

10B

1A1A

12A

15A

20B

20A

χ1

11

11

11

11

11

11

11

11

11

11

11

11

χ2

226

−2

4−

62

2−

32

2−

20

10

00

1−

20

00

−1

−1

−1

χ3

7713

15

55

12

−3

21

10

1−

1−

1−

21

00

−1

00

0

χ4

154

1010

1−

26

−2

44

−1

11

00

00

00

00

11

−2

−2

χ5

154

10−

101

−10

−2

24

4−

1−

11

00

2−

20

00

0−

11

00

χ6

154

10−

101

−10

−2

24

4−

1−

11

00

−2

20

00

0−

11

00

χ7

175

1511

415

−1

30

50

20

0−

11

10

1−

1−

10

−1

00

χ8

231

7−

96

15−

1−

16

11

0−

20

−1

−1

−1

21

00

01

00

χ9

693

219

021

51

−7

3−

20

00

1−

1−

11

−1

00

00

11

χ10

770

34−

105

−14

2−

2−

50

0−

11

0−

20

0−

10

00

10

11

χ11

770

−14

105

−10

−2

−2

−5

00

11

00

00

10

00

−1

0a5

a5

χ12

770

−14

105

−10

−2

−2

−5

00

11

00

00

10

00

−1

0a5

a5

χ13

825

259

6−

151

10

−5

00

−2

−1

11

10

−1

00

01

00

χ14

896

016

−4

00

0−

41

1−

20

00

00

01

b11

b11

01

00

χ15

896

016

−4

00

0−

41

1−

20

00

00

01

b11

b11

01

00

χ16

1056

320

−6

00

06

−4

10

2−

10

00

20

00

0−

10

0

χ17

1386

−6

180

6−

2−

211

61

00

00

00

−1

−2

00

00

11

χ18

1408

016

40

00

8−

7−

2−

20

10

00

01

00

0−

10

0

χ19

1750

−10

10−

5−

106

20

00

1−

10

−2

00

00

11

−1

00

0

χ20

1925

5−

19−

15

5−

30

50

−1

−1

01

11

01

00

−1

−1

00

χ21

1925

51

−1

−35

−3

10

50

1−

10

1−

1−

10

10

01

−1

00


[g]

1A2A

2B3A

4A4B

4C5A

5B5C

6A6B

7A8A

8B8C

10A

10B

11A

11B

12A

15A

20A

20B

χ22

2520

240

024

−8

0−

50

00

00

00

0−

10

11

00

−1

−1

χ23

2750

−50

−10

510

22

00

0−

11

−1

00

00

00

01

00

0

χ24

3200

0−

16−

40

00

0−

50

20

10

00

0−

1−

1−

10

10

0

Tabl

e 11

con

tinu

ed


Table 12 Character table of U6(2) - Part I

[g] 1A 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 5A

[g2] 1A 1A 1A 1A 3A 3B 3C 2A 2A 2B 2B 2B 2B 2B 5A

[g3] 1A 2A 2B 2C 1A 1A 1A 4A 4B 4C 4D 4E 4F 4G 5A

[g5] 1A 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 1A



χ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

χ2 22 −10 6 −2 4 −5 4 6 −2 2 2 2 −2 2 2

χ3 231 39 7 −9 6 15 6 23 7 −1 −1 −1 −1 −1 1

χ4 252 60 28 12 9 9 9 12 −4 4 4 4 4 4 2

χ5 385 −95 17 −7 25 7 −2 1 9 5 5 5 −7 −3 0

χ6 440 120 24 8 35 8 −1 24 8 0 0 0 8 0 0

χ7 560 −80 −16 16 20 20 2 16 −16 0 0 0 0 0 0

χ8 616 −24 40 8 −14 −5 13 8 8 8 8 8 0 0 1

χ9 770 −30 −14 10 5 −13 5 34 −6 −2 −2 −2 2 −2 0

χ10 770 −30 −14 10 5 −13 5 34 −6 −2 −2 −2 2 −2 0

χ11 1155 195 35 19 30 −6 3 −13 3 11 −5 −5 3 3 0

χ12 1155 195 35 19 30 −6 3 −13 3 −5 11 −5 3 3 0

χ13 1155 195 35 19 30 −6 3 −13 3 −5 −5 11 3 3 0

χ14 1386 −246 58 −30 36 9 9 −6 2 −2 −2 −2 −6 6 1

χ15 1540 260 4 −28 55 1 1 −12 −12 4 4 4 4 −4 0

χ16 3080 −440 40 −8 65 2 −7 40 −8 0 0 0 −8 0 0

χ17 3080 −440 40 −8 65 2 −7 40 −8 0 0 0 −8 0 0

χ18 3520 −320 64 0 10 −44 10 64 0 0 0 0 0 0 0

χ19 4620 −180 44 −36 −15 57 12 28 −20 4 4 4 −4 −4 0

χ20 4928 320 64 64 −4 68 5 0 0 0 0 0 0 0 −2

χ21 5544 −24 −56 24 9 36 9 72 24 0 0 0 −8 0 −1

χ22 6160 400 −48 −16 40 −50 4 48 16 0 0 0 0 0 0

χ23 6160 400 −48 −16 40 −50 4 48 16 0 0 0 0 0 0

χ24 6930 690 98 42 45 −36 −9 18 −6 6 6 6 10 −2 0

χ25 8064 384 128 0 −36 −36 18 0 0 0 0 0 0 0 −1

χ26 9240 −360 88 −8 −30 33 −3 −8 −8 24 −8 −8 0 0 0

χ27 9240 −360 88 −8 −30 33 −3 −8 −8 −8 24 −8 0 0 0

χ28 9240 −360 88 −8 −30 33 −3 −8 −8 −8 −8 24 0 0 0

χ29 10395 315 −21 −45 0 27 0 75 −13 −1 −1 −1 3 −1 0

χ30 10395 315 −21 −45 0 27 0 75 −13 −1 −1 −1 3 −1 0

χ31 10395 315 −21 −45 0 27 0 −21 19 15 −1 −1 3 −1 0

χ32 10395 315 −21 −45 0 27 0 −21 19 −1 15 −1 3 −1 0

χ33 10395 315 −21 −45 0 27 0 −21 19 −1 −1 15 3 −1 0

χ34 11264 1024 0 0 104 32 −4 0 0 0 0 0 0 0 −1

χ35 13860 420 100 36 −45 9 −18 84 20 4 4 4 −4 4 0

χ36 14784 −1344 64 0 114 −12 6 −64 0 0 0 0 0 0 −1

χ37 18711 −1161 −57 63 81 0 0 −9 15 3 3 3 −1 −5 1

χ38 18711 1431 87 −9 81 0 0 −9 −9 −9 −9 −9 −1 −1 1

χ39 20790 −810 6 −18 0 54 0 6 14 −6 −6 −6 6 2 0

χ40 20790 −810 6 −18 0 54 0 6 14 −6 −6 −6 6 2 0

χ41 24640 −960 64 −64 −20 −92 −11 0 0 0 0 0 0 0 0

χ42 25515 −405 −117 27 0 0 0 27 −21 3 3 3 3 3 0

χ43 25515 −405 −117 27 0 0 0 27 −21 3 3 3 3 3 0

χ44 32768 0 0 0 −64 −64 8 0 0 0 0 0 0 0 −2

χ45 37422 270 30 54 −81 0 0 −18 6 −6 −6 −6 −2 −6 2

χ46 40095 1215 −81 −9 0 0 0 −81 −9 3 3 3 −9 3 0


Table 13 Character table of U6(2) - Part II. aq = q+ 6i√3, c−3 = (1− 3i

√3)/2

[g] 6A 6B 6C 6D 6E 6F 6G 6H 7A 8A 8B 8C 8D 9A 9B 9C

[g2] 3B 3B 3A 3B 3A 3C 3C 3C 7A 4B 4C 4D 4E 9B 9A 9C

[g3] 2A 2A 2A 2B 2B 2A 2B 2C 7A 8A 8B 8C 8D 3B 3B 3B

[g5] 6B 6A 6C 6D 6E 6F 6G 6H 7A 8A 8B 8C 8D 9B 9A 9C

[g7] 6A 6B 6C 6D 6E 6F 6G 6H 1A 8A 8B 8C 8D 9A 9B 9C

[g11] 6B 6A 6C 6D 6E 6F 6G 6H 7A 8A 8B 8C 8D 9B 9A 9C

χ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

χ2 −1 −1 −4 3 0 2 0 −2 1 2 0 0 0 1 1 1

χ3 3 3 6 7 −2 0 −2 0 0 3 −1 −1 −1 0 0 0

χ4 −3 −3 9 1 1 3 1 3 0 0 0 0 0 0 0 0

χ5 −5 −5 1 −1 5 −2 2 2 0 1 −1 −1 −1 1 1 1

χ6 12 12 3 0 3 3 3 −1 −1 0 0 0 0 2 2 −1

χ7 −8 −8 4 −4 −4 −2 2 −2 0 0 0 0 0 −1 −1 2

χ8 3 3 −6 −5 −2 3 1 −1 0 0 0 0 0 −2 −2 −2

χ9 a−3 a−3 −3 7 1 −3 1 1 0 2 0 0 0 −1 −1 −1

χ10 a−3 a−3 −3 7 1 −3 1 1 0 2 0 0 0 −1 −1 −1

χ11 6 6 6 2 2 −3 −1 1 0 −1 3 −1 −1 0 0 0

χ12 6 6 6 2 2 −3 −1 1 0 −1 −1 3 −1 0 0 0

χ13 6 6 6 2 2 −3 −1 1 0 −1 −1 −1 3 0 0 0

χ14 −3 −3 −12 1 4 −3 1 −3 0 −2 0 0 0 0 0 0

χ15 17 17 −1 1 −5 −1 1 −1 0 0 0 0 0 1 1 1

χ16 a−8 a−8 1 −2 1 1 1 1 0 0 0 0 0 c−3 c−3 −1

χ17 a−8 a−8 1 −2 1 1 1 1 0 0 0 0 0 c−3 c−3 −1

χ18 4 4 −14 4 −2 4 −2 0 −1 0 0 0 0 1 1 1

χ19 9 9 9 −7 5 0 −4 0 0 0 0 0 0 0 0 0

χ20 −4 −4 −4 4 4 5 1 1 0 0 0 0 0 −1 −1 2

χ21 12 12 9 4 1 −3 1 −3 0 0 0 0 0 0 0 0

χ22 a4 a4 −8 −6 0 −2 0 2 0 0 0 0 0 −c−3 −c−3 1

χ23 a4 a4 −8 −6 0 −2 0 2 0 0 0 0 0 −c−3 −c−3 1

χ24 −12 −12 −3 −4 5 −3 −1 −3 0 2 0 0 0 0 0 0

χ25 −12 −12 12 −4 −4 0 2 0 0 0 0 0 0 0 0 0

χ26 9 9 −6 1 −2 −3 1 1 0 0 0 0 0 0 0 0

χ27 9 9 −6 1 −2 −3 1 1 0 0 0 0 0 0 0 0

χ28 9 9 −6 1 −2 −3 1 1 0 0 0 0 0 0 0 0

χ29 −9 −9 0 3 0 0 0 0 0 −1 1 1 1 0 0 0

χ30 −9 −9 0 3 0 0 0 0 0 −1 1 1 1 0 0 0

χ31 −9 −9 0 3 0 0 0 0 0 −1 −3 1 1 0 0 0

χ32 −9 −9 0 3 0 0 0 0 0 −1 1 −3 1 0 0 0

χ33 −9 −9 0 3 0 0 0 0 0 −1 1 1 −3 0 0 0

χ34 16 16 −8 0 0 4 0 0 1 0 0 0 0 −1 −1 −1

χ35 −3 −3 3 1 −5 0 −2 0 0 0 0 0 0 0 0 0

χ36 −12 −12 −6 4 −2 0 −2 0 0 0 0 0 0 0 0 0

χ37 0 0 9 0 −3 0 0 0 0 −1 1 1 1 0 0 0

χ38 0 0 9 0 −3 0 0 0 0 −1 −1 −1 −1 0 0 0

χ39 18i√3 −18i

√3 0 −6 0 0 0 0 0 2 0 0 0 0 0 0

χ40 −18i√3 18i

√3 0 −6 0 0 0 0 0 2 0 0 0 0 0 0

χ41 12 12 12 4 4 3 1 −1 0 0 0 0 0 −2 −2 1

χ42 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1 0 0 0

χ43 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1 0 0 0

χ44 0 0 0 0 0 0 0 0 1 0 0 0 0 2 2 −1


Table 13 continued

χ45 0 0 −9 0 3 0 0 0 0 −2 0 0 0 0 0 0

χ46 0 0 0 0 0 0 0 0 −1 3 1 1 1 0 0 0

Table 14 Character table of U6(2) - Part III. aq = q+ 6i√3, bp = (−1+ i

√p)/2,

dnm = m+ in

√3

[g] 10A 11A 11B 12A 12B 12C 12D 12E 12F 12G 12H 12I 15A 18A 18B

[g2] 5A 11B 11A 6B 6A 6C 6B 6A 6D 6D 6D 6E 15A 9B 9A

[g3] 10A 11A 11B 4A 4A 4A 4B 4B 4C 4D 4E 4F 5A 6A 6B

[g5] 2A 11A 11B 12B 12A 12C 12E 12D 12F 12G 12H 12I 3A 18B 18A

[g7] 10A 11B 11A 12A 12B 12C 12D 12E 12F 12G 12H 12I 15A 18A 18B

[g11] 10A 1A 1A 12B 12A 12C 12E 12D 12F 12G 12H 12I 15A 18B 18A

χ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

χ2 0 0 0 −3 −3 0 1 1 −1 −1 −1 −2 −1 −1 −1

χ3 −1 0 0 5 5 2 1 1 −1 −1 −1 2 1 0 0

χ4 0 −1 −1 3 3 −3 −1 −1 1 1 1 1 −1 0 0

χ5 0 0 0 1 1 1 −3 −3 −1 −1 −1 −1 0 1 1

χ6 0 0 0 6 6 3 2 2 0 0 0 −1 0 0 0

χ7 0 −1 −1 −2 −2 4 2 2 0 0 0 0 0 1 1

χ8 1 0 0 −1 −1 2 −1 −1 −1 −1 −1 0 1 0 0

χ9 0 0 0 d−3

−2d−3

−21 d−1

0d10

1 1 1 −1 0 d10 d−1

0

χ10 0 0 0 d−3

−2d−3

−21 d1

0 d−1

01 1 1 −1 0 d−1

0d10

χ11 0 0 0 −4 −4 2 0 0 2 −2 −2 0 0 0 0

χ12 0 0 0 −4 −4 2 0 0 −2 2 −2 0 0 0 0

χ13 0 0 0 −4 −4 2 0 0 −2 −2 2 0 0 0 0

χ14 −1 0 0 3 3 0 −1 −1 1 1 1 0 1 0 0

χ15 0 0 0 −3 −3 3 −3 −3 1 1 1 1 0 −1 −1

χ16 0 0 0 d−3

−5d−3

−51 d−1

1d−1

10 0 0 1 0 d

−1/2−1/2 d

−1/2−1/2

χ17 0 0 0 d−3

−5d−3

−51 d−1

1d−1

10 0 0 1 0 d

−1/2−1/2 d

−1/2−1/2

χ18 0 0 0 −8 −8 −2 0 0 0 0 0 0 0 1 1

χ19 0 0 0 1 1 1 1 1 1 1 1 −1 0 0 0

χ20 0 0 0 0 0 0 0 0 0 0 0 0 1 −1 −1

χ21 1 0 0 0 0 −3 0 0 0 0 0 1 −1 0 0

χ22 0 0 0 −6b3 −6b3 0 d−1

1d−1

10 0 0 0 0 d

−1/2−1/2 d

−1/2−1/2

χ23 0 0 0 −6b3 −6b3 0 d−1

1d−1

10 0 0 0 0 d

−1/2−1/2 d

−1/2−1/2

χ24 0 0 0 0 0 −3 0 0 0 0 0 1 0 0 0

χ25 −1 1 1 0 0 0 0 0 0 0 0 0 −1 0 0

χ26 0 0 0 1 1 −2 1 1 −3 1 1 0 0 0 0

χ27 0 0 0 1 1 −2 1 1 1 −3 1 0 0 0 0

χ28 0 0 0 1 1 −2 1 1 1 1 −3 0 0 0 0

χ29 0 0 0 −a−3 −a−3 0 d−2

−1d−2

−1−1 −1 −1 0 0 0 0

χ30 0 0 0 −a−3 −a−3 0 d−2

−1d−2

−1−1 −1 −1 0 0 0 0

χ31 0 0 0 −3 −3 0 1 1 3 −1 −1 0 0 0 0

χ32 0 0 0 −3 −3 0 1 1 −1 3 −1 0 0 0 0

χ33 0 0 0 −3 −3 0 1 1 −1 −1 3 0 0 0 0

χ34 −1 0 0 0 0 0 0 0 0 0 0 0 −1 1 1

χ35 0 0 0 3 3 3 −1 −1 1 1 1 −1 0 0 0

χ36 1 0 0 8 8 2 0 0 0 0 0 0 −1 0 0

χ37 −1 0 0 0 0 −3 0 0 0 0 0 −1 1 0 0


Table 14 continued

χ38 1 0 0 0 0 −3 0 0 0 0 0 −1 1 0 0

χ39 0 0 0 6b3 6b3 0 d1−1 d1−10 0 0 0 0 0 0

χ40 0 0 0 6b3 6b3 0 d1−1d1−1

0 0 0 0 0 0 0

χ41 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

χ42 0 −b11 −b11 0 0 0 0 0 0 0 0 0 0 0 0

χ43 0 −b11 −b11 0 0 0 0 0 0 0 0 0 0 0 0

χ44 0 −1 −1 0 0 0 0 0 0 0 0 0 1 0 0

χ45 0 0 0 0 0 3 0 0 0 0 0 1 −1 0 0

χ46 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Table 15 The twined series for M22. The table shows the Fourier coefficients multiplying q−D/12 in the q-expansion of the function hg,1(τ)

−D[g] 1A 2A 3A 4AB 5A 6A 7AB 8A 11AB

−1 −2 −2 −2 −2 −2 −2 −2 −2 −2

11 560 −16 2 0 0 2 0 0 −1

23 8470 54 −8 −2 0 0 0 2 0

35 70576 −144 16 0 −4 0 2 0 0

47 435820 332 4 4 0 −4 0 0 0

59 2187328 −704 −50 0 8 −2 −4 0 0

71 9493330 1394 58 −6 0 2 0 −2 0

83 36792560 −2640 38 0 0 6 0 0 2

95 130399766 4822 −172 6 −14 4 0 −2 2

107 429229920 −8480 174 0 0 −2 0 0 −2

119 1327987562 14506 104 −6 22 −8 6 2 0

131 3895785632 −24288 −502 0 −8 −6 4 0 2

143 10912966810 39770 466 10 0 2 −10 2 −2

155 29351354032 −63888 316 0 −28 12 0 0 0

167 76141761850 101018 −1232 −14 0 8 0 −2 0

179 191223891936 −157344 1098 0 56 −6 0 0 0

191 466389602756 241764 710 12 −14 −18 0 0 0

203 1107626293840 −366960 −2876 0 0 −12 10 0 0

215 2567229121428 550676 2472 −12 −62 8 6 4 4

227 5818567673360 −817840 1658 0 0 26 −20 0 2

239 12917927405852 1203068 −6148 20 112 20 0 0 −4

251 28135083779792 −1753840 5174 0 −48 −10 −4 0 0

263 60194978116000 2535488 3382 −24 0 −34 0 −4 2

275 126660856741328 −3637232 −12634 0 −112 −26 0 0 −3

287 262393981258310 5179526 10430 22 0 14 20 −2 0

Appendix C: Coefficient tablesSee Tables 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28.


Tabl

e 16

The

twin

ed s

erie

s fo

r M22. T

he ta

ble

show

s th

e Fo

urie

r coe

ffici

ents

mul

tipl

ying

q−D/12 in

the q-

expa

nsio

n of

the

func

tion

hg,2(τ)

−D

[g]

1A2A

3A4A

B5A

6A7A

B8A

11A

B

−4

11

11

11

11

1

821

02

3−

20

−1

00

1

2044

44−

47

4−

1−

1−

10

0

3242

560

16−

194

01

0−

21

4428

1512

−24

19−

87

30

00

5614

8196

412

24−

12−

60

10

0

6866

4920

0−

16−

8116

0−

15

0−

3

8026

4552

6464

7024

−6

−2

−4

40

9295

7314

05−

8370

−27

0−

20

10

104

3206

2637

236

−24

8−

3622

0−

40

0

116

1006

5671

56−

4421

744

−14

10

01

128

2990

3386

8016

818

352

03

0−

6−

1

140

8469

1294

48−

216

−65

6−

72−

170

30

−1

152

2300

0871

960

8855

8−

880

−2

100

0

164

6018

6506

768

−11

243

111

248

−1

−11

03

176

1523

3580

3872

416

−15

8214

4−

282

08

−1

188

3741

7253

0930

−49

413

40−

166

04

−5

−2

0

200

8943

5292

9498

202

981

−20

2−

421

00

−5

212

2085

1575

2830

0−

244

−35

4524

40

−1

00

0

224

4751

6756

0102

486

429

4628

010

4−

69

−12

0

236

1060

2363

9451

84−

1056

2077

−35

2−

51−

318

00

248

2319

9658

8165

8042

0−

7480

−42

00

0−

200

4

260

4985

1590

6540

96−

496

6146

496

−84

20

00

272

1053

2310

8387

200

1792

4228

608

04

−10

160

284

2190

2185

0730

991

−20

97−

1509

9−

697

196

−3

03

0


Tabl

e 17

The

twin

ed s

erie

s fo

r U4(3).

The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/12 in

the q-

expa

nsio

n of

the

func

tion

hg,1(τ)

−D

[g]

1A2A

3A3B

CD4A

B5A

6A6B

C7A

B8A

9ABC

D12

A

−1

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

−2

1156

0−

16−

342

00

22

00

−1

0

2384

7054

−11

6−

8−

20

00

02

1−

2

3570

576

−14

4−

272

160

−4

00

20

10

4743

5820

332

−66

24

40

2−

40

0−

2−

2

5921

8732

8−

704

−14

54−

500

8−

2−

2−

40

40

7194

9333

013

94−

2732

58−

60

−4

20

−2

10

8336

7925

60−

2640

−52

5438

00

66

00

−7

0

9513

0399

766

4822

−98

02−

172

6−

1410

40

−2

20

107

4292

2992

0−

8480

−16

782

174

00

−2

−2

00

30

119

1327

9875

6214

506

−28

930

104

−6

22−

14−

86

2−

70

131

3895

7856

32−

2428

8−

4906

6−

502

0−

8−

6−

64

05

0

143

1091

2966

810

3977

0−

7905

846

610

014

2−

102

4−

2

155

2935

1354

032

−63

888

−12

7484

316

0−

2812

120

0−

110

167

7614

1761

850

1010

18−

2032

64−

1232

−14

0−

48

0−

213

−2

179

1912

2389

1936

−15

7344

−31

3578

1098

056

−6

−6

00

30

191

4663

8960

2756

2417

64−

4828

4271

012

−14

−6

−18

00

−22

0

203

1107

6262

9384

0−

3669

60−

7367

72−

2876

00

−12

−12

100

100

215

2567

2291

2142

855

0676

−10

9887

624

72−

12−

62−

48

64

90

227

5818

5676

7336

0−

8178

40−

1634

074

1658

00

2626

−20

0−

220

239

1291

7927

4058

5212

0306

8−

2412

226

−61

4820

112

3820

00

172

251

2813

5083

7797

92−

1753

840

−35

0248

651

740

−48

−10

−10

−4

014

0

263

6019

4978

1160

0025

3548

8−

5067

686

3382

−24

0−

58−

340

−4

−32

0

275

1266

6085

6741

328

−36

3723

2−

7287

046

−12

634

0−

112

−26

−26

00

320

287

2623

9398

1258

310

5179

526

−10

3485

7010

430

220

3814

20−

211

−2


Tabl

e 18

The

twin

ed s

erie

s fo

r U4(3).

The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/12 in

the q-

expa

nsio

n of

the

func

tion

hg,2(τ)

−D

[g]

1A2A

3A3B

CD4A

B5A

6A6B

C7A

B8A

9ABC

D12

A

−4

11

11

11

11

11

11

821

02

213

−2

05

−1

00

01

2044

44−

497

74

−1

−7

−1

−1

01

1

3242

560

1619

7−

194

013

10

−2

−1

1

4428

1512

−24

577

19−

87

−15

30

0−

21

5614

8196

412

1176

24−

12−

624

01

03

0

6866

4920

0−

1623

13−

8116

0−

31−

15

0−

31

8026

4552

6464

4552

7024

−6

40−

2−

44

−2

0

9295

7314

05−

8384

4070

−27

0−

56−

20

17

0

104

3206

2637

236

1444

0−

248

−36

2272

0−

40

−5

0

116

1006

5671

56−

4425

759

217

44−

14−

891

00

1−

1

128

2990

3386

8016

842

861

183

520

117

30

−6

91

140

8469

1294

48−

216

6990

4−

656

−72

−17

−14

40

30

−5

0

152

2300

0871

960

8811

4066

558

−88

017

8−

210

0−

92

164

6018

6506

768

−11

218

1097

431

112

48−

223

−1

−11

011

1

176

1523

3580

3872

416

2802

08−

1582

144

−28

272

20

8−

100

188

3741

7253

0930

−49

443

6490

1340

−16

60

−32

64

−5

−2

−4

2

200

8943

5292

9498

202

6624

9998

1−

202

−42

403

10

024

−1

212

2085

1575

2830

0−

244

9930

25−

3545

244

0−

487

−1

00

−17

1

224

4751

6756

0102

486

414

8546

229

4628

010

458

2−

69

−12

−6

−2

236

1060

2363

9451

84−

1056

2189

455

2077

−35

2−

51−

705

−3

180

25−

1

248

2319

9658

8165

8042

031

8644

0−

7480

−42

00

840

0−

200

−16

0

260

4985

1590

6540

96−

496

4635

278

6146

496

−84

−99

42

00

−16

−2

272

1053

2310

8387

200

1792

6654

706

4228

608

011

864

−10

1634

2

284

2190

2185

0730

991

−20

9794

7538

3−

1509

9−

697

196

−14

01−

30

3−

30−

1


Tabl

e 19

The

twin

ed s

erie

s fo

r M23. T

he ta

ble

show

s th

e Fo

urie

r coe

ffici

ents

mul

tipl

ying

q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,1 2

(τ) .b

p=

(−1+

i√p)/2

−D

[g]

1A2A

3A4A

5A6A

7AB

8A11

AB

14A

B15

AB

23A

23B

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

2377

0−

145

−2

01

00

00

0b23

b23

4713

915

4310

−1

0−

2−

11

01

00

0

7113

2825

−11

921

5−

51

0−

10

01

00

9591

5124

308

314

4−

10

01

01

00

119

5069

867

−69

359

−13

23

−2

−1

00

−1

00

143

2405

3215

1407

85−

90

−3

4−

1−

10

0−

1−

1

167

1012

6854

0−

2772

135

240

32

2−

10

00

0

191

3877

4628

253

0620

014

−18

−4

00

00

00

0

215

1372

9350

90−

9710

300

−38

154

−5

2−

1−

10

00

239

4552

0392

9617

136

414

−20

11−

60

22

0−

10

0

263

1426

5412

315

−29

589

610

630

60

−3

00

01

1

287

4256

8680

715

5015

583

535

0−

5−

6−

10

00

00

311

1216

6594

9240

−83

160

1165

−10

8−

459

10−

20

00

00

335

3346

5824

6604

1351

4815

81−

6034

−11

3−

40

−1

10

0

359

8894

1309

5662

−21

6482

2158

170

2210

04

30

−2

00

383

2291

4821

4883

534

2259

2865

870

−11

−9

10

10

00

407

5739

3332

2767

0−

5336

1038

55−

250

015

02

−2

00

00

431

1400

8317

4239

6882

1296

5051

−12

4−

107

−17

06

−1

01

−1

−1

455

3338

8385

2016

99−

1250

717

6656

371

7916

−11

−5

01

10

0

479

7785

3744

7689

0618

8623

486

4919

056

−19

220

−4

0−

11

1

503

1778

8179

4535

250

−28

1679

811

250

−55

40

228

−4

32

00

0

527

3988

0841

9854

845

4167

709

1443

0−

283

0−

260

−7

00

01

1

551

8784

6157

5586

727

−61

1666

518

581

799

−21

829

−22

70

−2

10

0

575

1903

2414

7816

7799

8909

383

2363

139

515

4−

290

10

01

b23

b23


Tabl

e 20

The

twin

ed s

erie

s fo

r M23. T

he ta

ble

show

s th

e Fo

urie

r coe

ffici

ents

mul

tipl

ying

q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,3 2

(τ)

−D

[g]

1A2A

3A4A

5A6A

7AB

8A11

AB

14A

B15

AB

23A

B

−9

11

11

11

11

11

11

1523

17

6−

11

−2

0−

10

01

1

3957

96−

2818

41

20

0−

10

−2

0

6365

505

97−

151

01

−1

10

−1

01

8749

4385

−23

960

−7

04

31

1−

10

0

111

2922

381

525

90−

3−

9−

60

10

00

1

135

1452

5511

−11

13−

7515

6−

30

−1

00

0−

1

159

6344

7087

2255

228

77

−4

−3

−1

01

−2

0

183

2501

8843

5−

4333

360

−29

08

0−

12

00

1

207

9078

7658

579

45−

260

−15

04

0−

30

00

0

231

3073

1558

10−

1417

476

250

−25

10−

32

−1

12

0

255

9804

6607

7724

809

1062

2517

−10

91

−1

12

0

279

2971

7775

186

−42

286

−75

9−

7816

−7

12

01

10

303

8611

6649

220

7030

820

70−

360

−18

04

−3

00

0

327

2398

0659

2730

−11

5046

2880

122

016

−9

−2

1−

10

−1

351

6444

1843

4331

1855

63−

1926

59−

6410

0−

11

0−

10

375

1676

9940

6590

1−

2944

8352

56−

195

4624

0−

30

01

−1

399

4238

7885

8498

746

0571

6948

−10

137

−28

−9

−5

0−

1−

20

423

1043

2762

5252

95−

7119

85−

4645

295

0−

1319

30

−1

00

447

2505

8448

4337

7010

8884

212

120

146

0−

323

24

−1

00

471

5884

8028

3092

24−

1647

128

1591

2−

424

−13

640

04

00

20

495

1353

4935

1964

727

2466

359

−10

281

−20

192

23−

167

−1

0−

10

519

3053

2688

0332

593

−36

6033

526

646

617

6846

0−

7−

20

−4

0

543

6764

3308

3819

185

5388

145

3411

030

50

−50

0−

30

00

0

567

1473

4684

2984

7035

−78

6739

7−

2184

0−

901

0−

32−

15−

5−

51

01


Tabl

e 21

The

twin

ed s

erie

s fo

r McL

. The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,1 2

(τ) . bp=

(−1+

i√p)/2

−D

[g]

1A2A

3A3B

4A5A

5B6A

6B7A

B8A

9AB

10A

11A

B12

A14

AB

15A,15B

30A,30B

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

2377

0−

14−

135

−2

−5

07

10

0−

11

01

0b15

b15

4713

915

43−

1710

−1

−10

07

−2

−1

11

−2

0−

11

−b15

b15

7113

2825

−11

9−

4221

5−

25−

522

10

−1

01

02

0−b15

b15

9591

5124

308

−68

314

−26

432

−1

00

−2

−2

1−

20

b15

b15

119

5069

867

−69

3−

112

59−

13−

582

603

−2

−1

22

02

0−b15

(b15−

2)

143

2405

3215

1407

−16

785

−9

−85

081

−3

4−

11

−3

−1

−3

0−b15

(b15−

1)

167

1012

6854

0−

2772

−27

913

524

−13

50

141

32

2−

33

−1

30

(b15−

1)

(b15−

1)

191

3877

4628

253

06−

394

200

14−

218

−18

194

−4

00

2−

40

−4

0−2b15

2b15

215

1372

9350

90−

9710

−60

030

0−

38−

285

1530

44

−5

20

5−

14

−1

02b15

239

4552

0392

9617

136

−83

741

4−

20−

404

1141

1−

60

2−

3−

42

−5

0(2b15−

1)

(2b15−

3)

263

1426

5412

315

−29

589

−12

0861

063

−61

00

612

60

−3

46

06

02

(2b15−

2)

287

4256

8680

715

5015

5−

1667

835

35−

835

082

9−

5−

6−

11

−5

0−

70

−b15

(3b15−

2)

311

1216

6594

9240

−83

160

−23

4511

65−

108

−12

10−

4511

799

10−

2−

510

09

0(b

15−

2)

(3b15−

2)

335

3346

5824

6604

1351

48−

3153

1581

−60

−15

4634

1567

−11

3−

43

−12

0−

9−

1(1

−2b15)

(4b15−

1)

359

8894

1309

5662

−21

6482

−43

1321

5817

0−

2138

2221

6710

04

18

311

0b15

(5b15−

3)

383

2291

4821

4883

534

2259

−57

4828

6587

−28

650

2860

−11

−9

1−

6−

110

−12

1(4b15−

1)

(4b15−

3)

407

5739

3332

2767

0−

5336

10−

7692

3855

−25

0−

3855

038

6415

02

615

−2

140

(2−

2b15)

(4b15−

4)

431

1400

8317

4239

6882

1296

−10

096

5051

−12

4−

5157

−10

750

32−

170

62

−19

−1

−16

0(1

−b15)

(5b15−

3)

455

3338

8385

2016

99−

1250

717

−13

333

6656

371

−65

7679

6679

16−

11−

5−

718

017

1(4b15−

1)

(6b15−

3)

479

7785

3744

7689

0618

8623

4−

1728

086

4919

0−

8594

5686

24−

1922

06

−16

−4

−20

0(1

−3b15)

(7b15−

5)

503

1778

8179

4535

250

−28

1679

8−

2250

011

250

−55

4−

1125

00

1127

222

8−

40

223

222

0(8b15−

4)

527

3988

0841

9854

845

4167

709

−28

887

1443

0−

283

−14

430

014

413

−26

0−

7−

9−

260

−25

0(3

−3b15)

(9b15−

5)

551

8784

6157

5586

727

−61

1666

5−

3713

818

581

799

−18

798

−21

818

602

29−

227

830

028

−2

(2−

5b15)

(9b15−

6)

575

1903

2414

7816

7799

8909

383

−47

253

2363

139

5−

2347

615

423

599

−29

01

3−

320

−31

02

(10b15−

6)


Tabl

e 22

The

twin

ed s

erie

s fo

r McL

. The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,3 2

(τ)

−D

[g]

1A2A

3A3B

4A5A

5B6A

6B7A

B8A

9AB

10A

11A

B12

A14

AB

15A

B30

AB

−9

11

11

11

11

11

11

11

11

11

1523

17

156

−1

61

7−

20

−1

02

0−

10

02

3957

96−

2845

184

211

52

00

0−

3−

11

00

0

6365

505

9730

−15

130

022

1−

11

02

0−

2−

10

2

8749

4385

−23

915

060

−7

600

224

31

0−

41

2−

10

2

111

2922

381

525

225

90−

381

−9

57−

60

10

50

−3

00

2

135

1452

5511

−11

1315

9−

7515

161

663

−3

0−

13

−3

03

0−

13

159

6344

7087

2255

570

228

723

77

122

−4

−3

−1

05

0−

21

02

183

2501

8843

5−

4333

900

360

−29

360

016

48

0−

10

−8

24

00

4

207

9078

7658

579

4550

5−

260

−15

510

028

94

0−

3−

510

0−

30

04

231

3073

1558

10−

1417

419

0576

250

735

−25

361

10−

32

0−

9−

15

10

6

255

9804

6607

7724

809

2655

1062

2510

7717

551

−10

91

09

−1

−5

10

6

279

2971

7775

186

−42

286

1518

−75

9−

7815

3616

710

−7

12

0−

160

61

35

303

8611

6649

220

7030

851

7520

70−

3620

700

1071

−18

04

018

−3

−9

00

6

327

2398

0659

2730

−11

5046

7200

2880

122

2880

014

0816

−9

−2

0−

161

8−

10

8

351

6444

1843

4331

1855

6338

79−

1926

5938

06−

6419

9910

0−

19

181

−13

0−

19

375

1676

9940

6590

1−

2944

8313

140

5256

−19

553

0146

2580

240

−3

0−

230

120

010

399

4238

7885

8498

746

0571

1737

069

48−

101

6987

3735

30−

28−

9−

50

310

−14

−1

010

423

1043

2762

5252

95−

7119

8592

60−

4645

295

9270

045

32−

1319

3−

10−

300

16−

10

12

447

2505

8448

4337

7010

8884

230

300

1212

014

612

120

061

24−

323

20

324

−16

−1

014

471

5884

8028

3092

24−

1647

128

3978

015

912

−42

415

774

−13

678

7640

04

0−

380

200

016

495

1353

4935

1964

727

2466

359

2056

2−

1028

1−

201

2065

292

1044

223

−16

70

44−

1−

180

−3

17

519

3053

2688

0332

593

−36

6033

566

615

2664

661

726

718

6813

231

460

−7

0−

50−

223

00

16

543

6764

3308

3819

185

5388

145

8527

534

110

305

3411

00

1715

5−

500

−3

050

0−

250

020

567

1473

4684

2984

7035

−78

6739

743

725

−21

840

−90

143

710

021

637

−32

−15

−5

15−

62−

529

10

22


Tabl

e 23

The

twin

ed s

erie

s fo

r HS.

The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,1 2

(τ) . ap=

i√p

−D

[g]

1A2A

2B3A

4A4B

C5A

5BC

6A6B

7A8A

BC10

A10

B11

AB

12A

15A

20A,20B

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

2377

0−

1410

5−

10−

2−

50

11

00

10

0−

10

a5

4713

915

43−

4510

−25

−1

−10

00

−2

−1

1−

20

02

00

7113

2825

−11

912

921

−35

5−

25−

53

10

−1

1−

10

11

a5

9591

5124

308

−30

031

−60

4−

264

−3

−1

00

−2

01

−3

10

119

5069

867

−69

366

759

−12

5−

13−

582

13

−2

−1

22

01

−1

0

143

2405

3215

1407

−14

2585

−18

5−

9−

850

−3

−3

4−

1−

30

−1

10

a5

167

1012

6854

0−

2772

2820

135

−24

024

−13

50

33

22

30

−1

−3

0a5

191

3877

4628

253

06−

5278

200

−39

414

−21

8−

18−

4−

40

0−

42

02

0(a

5+

1)

215

1372

9350

90−

9710

9634

300

−63

0−

38−

285

154

4−

52

5−

1−

10

0a5

239

4552

0392

9617

136

−17

176

414

−86

0−

20−

404

11−

4−

60

2−

4−

12

−2

−1

0

263

1426

5412

315

−29

589

2971

561

0−

1145

63−

610

06

60

−3

60

04

00

287

4256

8680

715

5015

5−

5008

583

5−

1645

35−

835

0−

9−

5−

6−

1−

50

0−

10

a5

311

1216

6594

9240

−83

160

8294

411

65−

2420

−10

8−

1210

−45

99

10−

210

−1

0−

50

0

335

3346

5824

6604

1351

48−

1352

6815

81−

3244

−60

−15

4634

−7

−11

3−

4−

122

05

1(a

5+

1)

359

8894

1309

5662

−21

6482

2168

2221

58−

4126

170

−21

3822

1210

04

82

32

−2

(a5−

1)

383

2291

4821

4883

534

2259

−34

2085

2865

−56

6587

−28

650

−13

−11

−9

1−

110

0−

70

a5

407

5739

3332

2767

0−

5336

1053

3110

3855

−79

30−

250

−38

550

1315

02

150

−2

50

a5

431

1400

8317

4239

6882

1296

−82

1544

5051

−10

260

−12

4−

5157

−10

7−

15−

170

6−

191

−1

31

a5

455

3338

8385

2016

99−

1250

717

1251

459

6656

−12

909

371

−65

7679

1816

−11

−5

18−

10

−6

1(a

5+

1)

479

7785

3744

7689

0618

8623

4−

1885

854

8649

−17

146

190

−85

9456

−21

−19

220

−16

−4

−4

5−

1(a

5−

1)

503

1778

8179

4535

250

−28

1679

828

1569

011

250

−23

010

−55

4−

1125

00

2222

8−

422

03

00

2a5

527

3988

0841

9854

845

4167

709

−41

6827

514

430

−29

195

−28

3−

1443

00

−24

−26

0−

7−

260

0−

80

0

551

8784

6157

5586

727

−61

1666

561

1826

318

581

−36

305

799

−18

798

−21

827

29−

227

30−

20

71

2a5

575

1903

2414

7816

7799

8909

383

−89

0859

323

631

−46

925

395

−23

476

154

−33

−29

01

−32

20

11

0


Tabl

e 24

The

twin

ed s

erie

s fo

r HS.

The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,3 2

(τ)

−D

[g]

1A2A

2B3A

4A4B

C5A

5BC

6A6B

7A8A

BC10

A10

B11

AB

12A

15A

20A

B

−9

11

11

11

11

11

11

11

11

11

1523

17

−9

615

−1

61

0−

20

−1

21

00

10

3957

96−

2836

1836

421

10

20

0−

31

−1

0−

21

6365

505

97−

95−

1565

130

0−

51

−1

12

00

−1

00

8749

4385

−23

922

560

105

−7

600

04

31

−4

01

00

0

111

2922

381

525

−53

190

189

−3

81−

90

−6

01

5−

10

00

−1

135

1452

5511

−11

1311

43−

7531

915

161

69

−3

0−

1−

3−

20

10

−1

159

6344

7087

2255

−22

4122

847

17

237

70

−4

−3

−1

5−

10

0−

21

183

2501

8843

5−

4333

4275

360

675

−29

360

00

80

−1

−8

02

00

0

207

9078

7658

579

45−

7975

−26

010

25−

1551

00

−10

40

−3

100

02

00

231

3073

1558

10−

1417

414

274

762

1554

5073

5−

250

10−

32

−9

−1

−1

02

−1

255

9804

6607

7724

809

−24

759

1062

2169

2510

7717

0−

109

19

1−

10

2−

1

279

2971

7775

186

−42

286

4213

0−

759

2930

−78

1536

1619

−7

12

−16

00

−1

10

303

8611

6649

220

7030

8−

7038

020

7041

40−

3620

700

0−

180

418

0−

30

00

327

2398

0659

2730

−11

5046

1152

9028

8058

5012

228

800

016

−9

−2

−16

01

00

0

351

6444

1843

4331

1855

63−

1854

45−

1926

7835

5938

06−

64−

3610

0−

118

01

−4

−1

0

375

1676

9940

6590

1−

2944

8329

4093

5256

1026

9−

195

5301

460

240

−3

−23

−2

00

1−

1

399

4238

7885

8498

746

0571

−46

0773

6948

1385

1−

101

6987

370

−28

−9

−5

31−

30

0−

21

423

1043

2762

5252

95−

7119

8571

2575

−46

4518

775

295

9270

045

−13

193

−30

00

10

0

447

2505

8448

4337

7010

8884

2−

1088

550

1212

024

450

146

1212

00

0−

323

232

04

00

0

471

5884

8028

3092

24−

1647

128

1646

280

1591

231

320

−42

415

774

−13

60

400

4−

380

00

20

495

1353

4935

1964

727

2466

359

−24

6676

1−

1028

141

015

−20

120

652

92−

5923

−16

744

4−

15

−1

0

519

3053

2688

0332

593

−36

6033

536

6156

926

646

5381

761

726

718

680

460

−7

−50

4−

20

−4

2

543

6764

3308

3819

185

5388

145

−53

8753

534

110

6862

530

534

110

00

−50

0−

350

00

00

0

567

1473

4684

2984

7035

−78

6739

778

6559

5−

2184

086

395

−90

143

710

090

−32

−15

−5

−62

0−

5−

20

0


Tabl

e 25

The

twin

ed s

erie

s fo

r U6(2).

The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,1 2

(τ) -

par

t I. a

p=

i√p

−D

[g]

1A2A

2B2C

3A3B

3C4A

4B4C

DE

4F4G

5A6A,6B

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

2377

0−

30−

1410

5−

135

34−

6−

22

−2

0−

3– 6a3

4713

915

−5

43−

4510

−17

1013

9−

13−

13

−1

0−

5

7113

2825

−19

9−

119

129

21−

4221

505

−15

5−

75

−5

−10−

12a3

9591

5124

180

308

−30

031

−68

3114

12−

284

−4

44

−18−

18a3

119

5069

867

−91

7−

693

667

59−

112

5936

27−

69−

1311

−13

2−

26−

30a3

143

2405

3215

1055

1407

−14

2585

−16

785

8559

−97

−9

7−

90

−43−36a3

167

1012

6854

0−

3300

−27

7228

2013

5−

279

135

1906

8−

108

24−

2024

0−

69−

78a3

191

3877

4628

244

9053

06−

5278

200

−39

420

040

154

−19

014

−14

14−

18−

100−

90a3

215

1372

9350

90−

1089

4−

9710

9634

300

−60

030

081

250

−33

4−

3842

−38

15−

148−

156a3

239

4552

0392

9615

456

1713

6−

1717

641

4−

837

414

1587

36−

440

−20

24−

2011

−21

3−204a3

263

1426

5412

315

−32

005

−29

589

2971

561

0−

1208

610

3009

87−

541

63−

6963

0−

298−

306a3

287

4256

8680

715

4679

550

155

−50

085

835

−16

6783

555

5307

−80

535

−37

350

−41

9− 408a3

311

1216

6594

9240

−87

784

−83

160

8294

411

65−

2345

1165

1001

016

−12

64−

108

104

−10

8−

45−

583−

606a3

335

3346

5824

6604

1287

8013

5148

−13

5268

1581

−31

5315

8117

6738

8−

1652

−60

52−

6034

−79

3− 768a3

359

8894

1309

5662

−22

5074

−21

6482

2168

2221

58−

4313

2158

3062

830

−19

7817

0−

162

170

22−

1073−

1092a3

383

2291

4821

4883

533

0755

3422

59−

3420

8528

65−

5748

2865

5217

427

−27

8987

−85

870

−14

44− 1

428a3

407

5739

3332

2767

0−

5489

70−

5336

1053

3110

3855

−76

9238

5587

5146

2−

4090

−25

025

4−

250

0−

1914−

1938a3

431

1400

8317

4239

6880

1024

8212

96−

8215

4450

51−

1009

650

5114

4731

68−

5192

−12

413

6−

124

−10

7−

2532−

2496a3

455

3338

8385

2016

99−

1277

277

−12

5071

712

5145

966

56−

1333

366

5623

6257

63−

6269

371

−38

137

179

−33

27−

3366a3

479

7785

3744

7689

0618

5156

218

8623

4−

1885

854

8649

−17

280

8649

3810

1658

−84

7819

0−

190

190

56−

4328−

4284a3

503

1778

8179

4535

250

−28

6171

0−

2816

798

2815

690

1125

0−

2250

011

250

6076

2514

−11

782

−55

454

6−

554

0−

5614−

5658a3

527

3988

0841

9854

845

4109

885

4167

709

−41

6827

514

430

−28

887

1443

095

8953

25−

1473

9−

283

269

−28

30

−72

37−

7194a3

551

8784

6157

5586

727

−61

9087

3−

6116

665

6118

263

1858

1−

3713

818

581

1498

7243

9−

1775

379

9−

785

799

−21

8−

9268−

9318a3

575

1903

2414

7816

7799

8814

743

8909

383

−89

0859

323

631

−47

253

2363

123

2094

167

−23

265

395

−39

339

515

4−

1182

7− 11766a3


Tabl

e 26

The

twin

ed s

erie

s fo

r U6(2).

The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,1 2

(τ) -

par

t II. ap=

i√p

−D

[g]

6C6D

6E6F

6G6H

7A8A

8BCD

9ABC

10A

11A

B12A,12B

12C

12D,12E

12FG

H12

I15

A18A,18B

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

−1

23−

37

1−

31

10

20

−1

00

−2+

3a3

1a3

1−

10

−a3

47−

147

−2

4−

20

−1

−1

11

00

−5+

6a3

−2

−1−

2a3

−1

00

1

71−

1922

1−

11

30

1−

10

10

−8+

12a3

10

2−

11

−1+

a3

95−

3332

−1

−3

−1

−3

00

0−

20

1−

19+

15a3

−1

−1+

a3

−2

−1

10

119

−53

603

13

1−

23

−1

2−

20

−27+

27a3

33+

a3

2−

1−

11+

a3

143

−91

81−

35

−3

−3

4−

5−

11

0−

1−

45+

42a3

-3−

1−2 a

3−

31

0−

1

167

−12

914

13

−9

33

24

2−

30

−1

−66+

69a3

33 a

33

10

−a3

191

−20

819

4−

48

−4

−4

0−

20

20

0−

103+

99a3

-4−

1−3 a

3−

4−

20

2

215

−29

230

44

−4

44

−5

62

01

−1

−14

6+ 150a3

42+

2 a3

40

0−

1+a3

239

−42

641

1−

60

−6

−4

0−

42

−3

12

−21

3+ 210a3

−6

−5−

2 a3

−5

0−

1−a3

263

−59

861

26

26

60

3−

34

00

−29

7+ 303a3

65+

a3

60

02

287

−84

582

9−

57

−5

−9

−6

−5

−1

10

0−

425+

414a3

−5

−1−

2 a3

−7

−1

0−

2+a3

311

−11

4711

799

−19

99

108

−2

−5

10

−57

6+ 585a3

92+

7 a3

9−

10

−1−

2 a3

335

−16

0315

67−

1117

−11

−7

3−

16−

43

00

−79

7+ 792a3

−11

−5−

8 a3

−9

11

2+a3

359

−21

3821

6710

−8

1012

014

41

−4

3−

1067+

1080a3

105+

4 a3

110

−2

−2+

a3

383

−28

8728

60−

11−

1−

11−

13−

9−

91

−6

00

−14

48+

1434a3

−11

−8−

2 a3

−12

−1

0−

1−a3

407

−38

2538

6415

−3

1513

018

26

0−

2−

1911+

1923a3

1511+

5 a3

14−

10

3+a3

431

−50

8550

32−

1721

−17

−15

0−

126

2−

1−

1−

2540+

2526a3

−17

−8−

10a3

−16

11

−3+

a3

455

−66

2466

7916

−30

1618

−11

11−

5−

73

0−

3314+

3333a3

164+

11a3

170

1−

3 a3

479

−86

8786

24−

1931

−19

−21

22−

180

62

−4

−43

42+

4320a3

−19

−6−

12a3

−20

−1

−1

4

503

−11

206

1127

222

−22

2222

818

−4

00

3−

5603+

5625a3

2211+

11a3

220

0−

4+2 a

3

527

−14

482

1441

3−

268

−26

−24

0−

35−

7−

90

0−

7244+

7221a3

−26

−18−

9 a3

−25

20

−1−

2 a3

551

−18

523

1860

229

−13

2927

−22

357

82

0−

9259+

9285a3

2919+

11a3

281

15+

a3

575

−23

689

2359

9−

2935

−29

−33

0−

251

3−

20

−11

846+

11811a3

−29

−12−

15a3

−31

−3

1−

4+a3


Tabl

e 27

The

twin

ed s

erie

s fo

r U6(2).

The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,3 2

(τ) -

par

t I

−D

[g]

1A2A

2B2C

3A3B

3C4A

4B4C

DE

4F4G

5A6A

B6C

6D

−9

11

11

11

11

11

11

11

11

1523

139

7−

96

156

237

−1

−1

−1

13

67

3957

9636

−28

3618

4518

8420

4−

44

19

185

6365

505

225

97−

95−

1530

−15

353

331

11

018

3322

8749

4385

−15

−23

922

560

150

6099

349

−7

9−

70

3060

22

111

2922

381

909

525

−53

190

225

9026

8593

−3

5−

3−

945

9057

135

1452

5511

−50

5−

1113

1143

−75

159

−75

6487

167

15−

1715

671

149

63

159

6344

7087

3183

2255

−22

4122

857

022

814

719

239

7−

97

711

422

812

2

183

2501

8843

5−

2925

−43

3342

7536

090

036

031

395

323

−29

27−

290

180

360

164

207

9078

7658

510

025

7945

−79

75−

260

505

−26

064

553

505

−15

9−

150

269

524

289

231

3073

1558

10−

1116

6−

1417

414

274

762

1905

762

1274

9080

250

−46

50−

2538

176

236

1

255

9804

6607

7729

097

2480

9−

2475

910

6226

5510

6224

3945

1097

25−

2325

1753

110

6255

1

279

2971

7775

186

−36

270

−42

286

4213

0−

759

1518

−75

945

3842

1426

−78

82−

7816

738

1497

710

303

8611

6649

220

7866

070

308

−70

380

2070

5175

2070

8245

9620

52−

3644

−36

010

3520

7010

71

327

2398

0659

2730

−10

3590

−11

5046

1152

9028

8072

0028

8014

6557

829

8612

2−

126

122

014

4028

8014

08

351

6444

1843

4331

2011

1518

5563

−18

5445

−19

2638

79−

1926

2555

051

3947

59−

6159

−64

1963

3898

1999

375

1676

9940

6590

1−

2735

55−

2944

8329

4093

5256

1314

052

5643

7607

750

37−

195

189

−19

546

2628

5256

2580

399

4238

7885

8498

748

8475

4605

71−

4607

7369

4817

370

6948

7377

627

6875

−10

191

−10

137

3474

6948

3530

423

1043

2762

5252

95−

6750

25−

7119

8571

2575

−46

4592

60−

4645

1225

7551

9535

295

−28

929

50

4592

9227

4532

447

2505

8448

4337

7011

3745

010

8884

2−

1088

550

1212

030

300

1212

020

0933

0612

298

146

−14

214

60

6060

1212

061

24

471

5884

8028

3092

24−

1583

640

−16

4712

816

4628

015

912

3978

015

912

3253

1448

1544

8−

424

432

−42

4−

136

7956

1591

278

76

495

1353

4935

1964

727

2548

791

2466

359

−24

6676

1−

1028

120

562

−10

281

5207

0711

2040

7−

201

215

−20

192

1035

020

631

1044

2

519

3053

2688

0332

593

−35

5393

5−

3660

335

3661

569

2664

666

615

2664

682

4581

4527

217

617

−63

161

768

1332

326

646

1323

1

543

6764

3308

3819

185

5524

785

5388

145

−53

8753

534

110

8527

534

110

1292

8092

934

465

305

−31

130

50

1705

534

110

1715

5

567

1473

4684

2984

7035

−76

9280

5−

7867

397

7865

595

−21

840

4372

5−

2184

020

0804

347

4274

7−

901

891

−90

10

2176

143

616

2163

7


Tabl

e 28

The

twin

ed s

erie

s fo

r U6(2).

The

tabl

e sh

ows

the

Four

ier c

oeffi

cien

ts m

ulti

plyi

ng q−D/24 in

the q-

expa

nsio

n of

the

func

tion

hg,3 2

(τ) -

par

t II

−D

[g]

6E6F

6G6H

7A8A

8BCD

9ABC

10A

11A

B12

AB

12C

12D

E12

FGH

12I

15A

18A

B

−9

11

11

11

11

11

11

11

11

1

15−

20

−2

00

3−

10

−1

05

21

−1

21

0

392

02

00

00

01

−1

3−

6−

11

2−

20

631

31

−5

−1

11

00

020

50

−2

10

0

874

04

03

−3

10

01

300

−2

20

00

111

−6

0−

60

09

10

−1

039

−6

3−

32

00

135

−3

−7

−3

90

−5

−1

30

079

1−

13

10

−1

159

−4

0−

40

−3

3−

10

30

130

162

−2

0−

20

183

80

80

0−

9−

10

02

156

−24

−4

40

00

207

414

4−

100

9−

3−

50

027

520

7−

30

0−

1

231

100

100

−3

−6

20

−1

−1

383

2−

55

22

0

255

−10

0−

100

99

10

−3

−1

513

−18

5−

5−

22

0

279

−7

−21

−7

191

−14

20

00

764

5−

86

11

0

303

−18

0−

180

024

40

0−

310

6934

9−

92

00

327

160

160

−9

−18

−2

00

113

68−

72−

88

00

0

351

1028

10−

360

15−

19

01

1985

505

−13

2−

11

375

240

240

0−

27−

30

00

2644

16−

1212

01

0

399

−28

0−

280

−9

27−

50

50

3414

−60

14−

144

−2

0

423

−13

−43

−13

4519

−25

3−

100

046

5419

−14

16−

10

2

447

−32

0−

320

330

20

04

6152

9216

−16

−4

00

471

400

400

0−

364

00

077

88−

168

−20

200

20

495

2369

23−

59−

1655

70

−4

−1

1040

011

928

−18

−1

−1

0

519

460

460

0−

51−

70

0−

213

357

34−

2323

2−

40

543

−50

0−

500

045

−3

00

016

917

−13

825

−25

−2

00

567

−32

−94

−32

90−

15−

69−

515

0−

521

907

52−

3329

00

−1


Tabl

e 29

The

tabl

e sh

ows

the

deco

mpo

siti

on o

f the

Fou

rier

coe

ffici

ents

mul

tipl

ying

q−D/12 in

the

func

tion

hg,1(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χ

n o

f M22

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

χ9

χ10

χ11

χ12

−1

−2

00

00

00

00

00

0

110

00

00

00

00

11

0

230

00

02

24

46

44

8

350

28

86

1622

3432

4848

60

472

2642

4260

100

160

208

232

268

268

380

592

9222

822

825

848

073

810

2811

3213

9413

9419

00

7126

472

952

952

1204

2130

3334

4508

4966

5966

5966

8240

8378

1710

3754

3754

4516

8192

1270

817

410

1911

223

284

2328

431

928

9530

062

2013

194

1319

416

254

2914

845

402

6175

268

016

8221

882

218

1132

16

107

950

2022

843

616

4361

653

078

9574

414

8822

2032

0422

3388

2711

6027

1160

3725

64

119

3040

6308

013

4624

1346

2416

4946

2965

3646

1482

6288

6869

1920

8380

7683

8076

1152

800

131

8706

1841

0039

5460

3954

6048

2650

8694

0813

5205

818

4442

820

2865

224

5995

224

5995

233

8172

4

143

2472

051

7270

1106

932

1106

932

1354

036

2436

238

3790

274

5167

368

5684

522

6888

684

6888

684

9473

140

155

6601

613

8893

029

7851

829

7851

836

3864

665

5114

810

1897

9413

8971

1415

2859

6818

5312

3618

5312

3625

4784

20

167

1719

0436

0640

877

2463

277

2463

294

4399

016

9967

1826

4407

0836

0523

9639

6591

1448

0671

5648

0671

5666

0956

08

179

4307

8290

5221

619

4030

0419

4030

0423

7104

4242

6826

6466

3931

0690

5408

8899

5928

2812

0725

462

1207

2546

216

5992

492

191

1052

210

2208

6048

4731

8486

4731

8486

5784

0438

1041

0671

416

1947

152

2208

2972

824

2915

594

2944

3286

229

4432

862

4048

5268

0

203

2496

320

5243

9410

1123

8378

011

2383

780

1373

4767

824

7235

148

3845

8282

652

4442

118

5768

8224

469

9266

588

6992

6658

896

1480

364

215

5789

798

1215

6177

026

0469

430

2604

6943

031

8366

892

5730

4653

089

1413

422

1215

5471

6813

3710

8322

1620

7140

5216

2071

4052

2228

4987

96

227

1311

6978

2754

8985

059

0364

314

5903

6431

472

1533

716

1298

7810

3220

2031

4748

2754

9995

1030

3048

8952

3673

3552

6436

7335

5264

5050

8377

68

239

2912

8906

6116

5946

613

1065

6938

1310

6569

3816

0194

7284

2883

4753

1644

8542

2600

6116

4486

4867

2810

9656

8155

2319

6881

5523

1968

1121

3482

124

App

endi

x D

: Dec

ompo

siti

on ta

bles

See

Tabl

es 2

9, 3

0, 3

1, 3

2, 3

3, 3

4, 3

5, 3

6, 3

7, 3

8, 3

9, 4

0, 4

1, 4

2, 4

3, 4

4, 4

5, 4

6, 4

7, 4

8, 4

9, 5

0, 5

1, 5

2, 5

3, 5

4, 5

5, 5

6, 5

7, 5

8, 5

9, 6

0.


Tabl

e 30

The

tabl

e sh

ows

the

deco

mpo

siti

on o

f the

Fou

rier

coe

ffici

ents

mul

tipl

ying

q−D/12 in

the

func

tion

hg,2(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χ

n o

f M22

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

χ9

χ10

χ11

χ12

−4

10

00

00

00

00

00

80

00

00

00

10

00

0

200

11

11

12

21

33

4

320

15

55

1015

1824

2626

38

442

1528

2835

6196

136

147

179

179

242

562

7014

914

918

433

251

670

677

093

693

612

84

6815

310

675

675

821

1486

2307

3138

3468

4196

4196

5778

8062

1260

2686

2686

3286

5906

9192

1252

813

770

1670

216

702

2296

4

9221

545

3597

1197

1111

871

2136

733

238

4533

849

855

6044

060

440

8309

4

104

716

1516

232

528

3252

839

754

7156

211

1316

1517

9816

7022

2024

0820

2408

2783

32

116

2277

4767

810

2132

1021

3212

4829

2246

8434

9513

4766

0452

4228

6354

6663

5466

8737

48

128

6752

1416

1130

3407

3034

0737

0835

6674

9010

3832

514

1588

615

5745

218

8784

218

8784

225

9577

4

140

1906

740

0933

8592

8185

9281

1050

214

1890

431

2940

645

4009

962

4411

059

5346

659

5346

659

7351

704

152

5187

010

8909

623

3368

223

3368

228

5230

251

3412

479

8642

610

8906

2811

9795

8414

5207

6614

5207

6619

9659

96

164

1357

3128

4979

461

0659

961

0659

974

6362

113

4344

7820

8980

9928

4974

1431

3471

0037

9965

4637

9965

4652

2452

18

176

3434

3672

1275

015

4561

5815

4561

5818

8908

1834

0035

4052

8943

5872

1285

3879

3416

8496

1715

4196

1715

4113

2236

044

188

8436

6217

7165

9237

9639

2337

9639

2346

4003

8883

5206

4412

9921

032

1771

6516

419

4881

422

2362

2008

023

6220

080

3248

0245

2

200

2016

488

4234

6314

9074

1956

9074

1956

1109

0688

619

9632

354

3105

3924

042

3462

681

4658

0875

056

4616

772

5646

1677

277

6347

956

212

4701

307

9872

8774

2115

6227

421

1562

274

2585

7598

746

5436

954

7240

1294

198

7290

154

1086

0198

1413

1638

7235

1316

3872

3518

1003

2796

224

1071

3686

2249

8494

648

2109

963

4821

0996

358

9245

588

1060

6418

7416

4988

7462

2249

8466

0024

7483

0808

2999

7952

6629

9979

5266

4124

7182

68

236

2390

5058

5020

0598

510

7572

6834

1075

7268

3413

1477

7297

2366

5990

9136

8137

6388

5020

0589

2055

2206

4385

6693

4116

6166

9341

1661

9203

4407

74

248

5230

7782

1098

4680

3623

5386

1448

2353

8614

4828

7694

1610

5178

4952

5680

5543

6866

1098

4686

398

1208

3156

304

1464

6249

036

1464

6249

036

2013

8593

100

260

1124

0007

023

6039

7820

5057

9942

3250

5799

4232

6181

9930

6611

1275

8722

817

3095

8029

423

6039

7326

425

9643

6954

831

4719

6397

831

4719

6397

843

2739

4992

4


Tabl

e 31

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/12 in

the

func

tion

hg,1(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U4(3) -

pa

rt I

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

χ9

χ10

−1

−2

00

00

00

00

0

110

00

00

00

00

0

230

00

00

00

00

0

350

00

00

24

47

7

470

64

414

2222

2634

34

590

1220

2052

8212

013

218

918

9

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7210

210

227

042

253

260

680

180

1

838

232

380

380

984

1548

2112

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3164

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860

1398

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8350

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511

125

107

124

2748

4554

4554

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306

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814

228

3667

657

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7667

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316

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0611

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241

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4162

210

7060

1666

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5288

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6033

4074

3340

74

143

3368

7044

611

6944

1169

4430

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4681

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1070

7015

1293

5210

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10

155

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1885

9231

4208

3142

0880

8100

1257

446

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156

1886

748

2516

637

2516

637

167

2335

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0188

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5984

2098

802

3264

774

4405

146

4895

338

6526

850

6526

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179

5838

812

2927

620

4845

020

4845

052

6778

881

9533

411

0651

4412

2944

2016

3949

4816

3949

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191

1429

3630

0048

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9813

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8538

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01

203

3386

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411

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8611

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5188

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4759

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8871

2174

1294

9623

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9623

20

215

7863

8216

5110

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5122

3027

5122

3070

7490

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0054

184

1485

5999

016

5070

996

2200

9286

922

0092

869

227

1780

628

3741

1832

6235

1670

6235

1670

1603

3454

424

9414

368

3367

1723

237

4129

064

4988

5130

949

8851

309

239

3955

942

8307

0266

1384

3786

613

8437

866

3559

9006

055

3762

164

7475

5092

483

0621

184

1107

4914

2711

0749

1427

251

8612

692

1809

0595

230

1506

846

3015

0684

677

5306

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1206

0439

8016

2817

6080

1809

0820

3224

1213

6251

2412

1362

51

263

1843

2618

3870

7255

064

5093

274

6450

9327

416

5882

6474

2580

3968

3034

8347

5608

3870

5481

6651

6072

2937

5160

7229

37

275

3877

7988

8144

2566

413

5737

1466

1357

3714

6634

9039

0608

5429

5186

6073

2988

6448

8144

3121

1210

8591

3941

710

8591

3941

7

287

8034

5934

1687

2376

7828

1200

6070

2812

0060

7072

3090

3886

1124

8072

954

1518

4775

484

1687

2013

154

2249

6001

167

2249

6001

167


Tabl

e 32

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/12 in

the

func

tion

hg,1(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U4(3) -

pa

rt II

−D

χ11

χ12

χ13

χ14

χ15

χ16

χ17

χ18

χ19

χ20

−1

00

00

00

00

00

110

00

00

10

00

0

230

02

22

12

22

2

357

76

610

1614

1416

16

4734

3446

4662

7486

8610

011

6

5918

918

920

620

629

039

143

243

248

059

6

7180

180

193

493

412

4216

2618

6418

6421

3025

86

8331

6431

6435

3235

3247

5663

7972

1672

1681

9210

064

9511

125

1112

512

636

1263

616

856

2234

525

568

2556

829

148

3572

8

107

3683

436

834

4134

641

346

5527

873

819

8413

684

136

9574

411

7656

119

1137

0611

3706

1282

7012

8270

1710

2222

7677

2602

7426

0274

2965

3636

4172

131

3340

7433

4074

3755

9037

5590

5012

5866

8609

7635

0076

3500

8694

0810

6856

0

143

9352

1093

5210

1053

068

1053

068

1404

072

1871

144

2138

648

2138

648

2436

238

2993

494

155

2516

637

2516

637

2830

546

2830

546

3775

230

5034

454

5751

956

5751

956

6551

148

8051

776

167

6526

850

6526

850

7345

188

7345

188

9793

632

1305

5603

1492

1266

1492

1266

1699

6718

2088

8326

179

1639

4948

1639

4948

1844

2654

1844

2654

2459

3074

3279

2777

3747

3264

3747

3264

4268

2664

5246

0200

191

3998

2901

3998

2901

4498

6624

4498

6624

5998

2140

7997

0261

9139

5836

9139

5836

1041

0671

412

7950

562

203

9496

2320

9496

2320

1068

2884

610

6828

846

1424

4537

818

9931

510

2170

5497

021

7054

970

2472

3514

830

3871

648

215

2200

9286

922

0092

869

2476

1781

224

7617

812

3301

5697

644

0195

866

5030

8382

650

3083

826

5730

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070

4309

074

227

4988

5130

949

8851

309

5611

9917

256

1199

172

7482

8063

699

7717

729

1140

2271

3611

4022

7136

1298

7810

3215

9630

5804

239

1107

4914

2711

0749

1427

1245

9572

2412

4595

7224

1661

2765

5622

1500

5306

2531

4411

9825

3144

1198

2883

4753

1635

4400

0144

251

2412

1362

5124

1213

6251

2713

6349

0427

1363

4904

3618

2121

3648

2430

4831

5513

4446

8055

1344

4680

6280

1389

3677

1879

6324

263

5160

7229

3751

6072

2937

5805

8748

6058

0587

4860

7741

1663

4010

3214

9273

111

7960

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611

7960

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613

4363

9885

216

5143

6929

2

275

1085

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417

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417

1221

6494

216

1221

6494

216

1628

8726

840

2171

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527

2482

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652

2482

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2827

2484

272

3474

9166

956

287

2249

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167

2249

6001

167

2530

8126

986

2530

8126

986

3374

4168

882

4499

2097

964

5141

9567

602

5141

9567

602

5857

0125

574

7198

7317

446


Tabl

e 33

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/12 in

the

func

tion

hg,2(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U4(3) -

pa

rt I

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

χ9

χ10

−4

10

00

00

00

00

80

00

00

00

10

0

200

00

01

11

01

1

320

01

12

14

33

3

441

24

48

1416

2126

26

560

818

1842

6492

104

129

129

683

3774

7418

528

539

642

857

357

3

8010

170

288

288

742

1142

1558

1720

2276

2276

9230

606

1038

1038

2648

4114

5574

6189

8226

8226

104

9820

4034

5834

5888

5213

740

1861

620

668

2750

527

505

116

315

6453

1082

010

820

2778

543

187

5837

464

812

8634

986

349

128

932

1920

032

105

3210

582

484

1282

3117

3260

1924

4325

6447

2564

47

140

2585

5434

390

862

9086

223

3470

3630

7249

0425

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9272

6201

7262

01

152

7068

1478

0224

6650

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4020

9860

9613

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414

7942

619

7215

719

7215

7

164

1847

938

6845

6452

4664

5246

1658

877

2580

201

3483

848

3870

656

5160

351

5160

351

176

4667

897

9182

1632

930

1632

930

4198

392

6530

326

8817

060

9796

194

1306

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1306

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188

1146

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200

2739

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5007

695

8545

695

8545

624

6469

9638

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7596

0457

5097

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6774

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6774

96

212

6385

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4064

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3474

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1340

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017

8770

283

1787

7028

3

224

1455

274

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5092

4466

5092

4466

1309

4583

020

3691

226

2749

8813

630

5540

174

4073

8219

040

7382

190

236

3246

731

6817

1414

1136

2573

011

3625

730

2921

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445

4492

898

6135

7280

868

1744

339

9089

8565

190

8985

651

248

7103

972

1491

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424

8629

024

2486

2902

463

9325

160

9945

0069

213

4258

7586

1491

7588

8019

8900

2072

1989

0020

72

260

1526

5066

3205

4326

653

4252

608

5342

5260

813

7378

3498

2136

9895

5828

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7397

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16

272

3225

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6772

2534

011

2872

8962

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7289

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0243

2966

4514

8853

9660

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6772

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5490

2977

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9029

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16

284

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4204

1408

3090

5323

4721

2278

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7860

3566

9290

9388

8040

8412

6749

1931

714

0832

2860

218

7776

0945

518

7776

0945

5


Tabl

e 34

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/12 in

the

func

tion

hg,1(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U4(3) -

pa

rt II

−D

χ11

χ12

χ13

χ14

χ15

χ16

χ17

χ18

χ19

χ20

−4

00

00

00

00

00

80

00

00

00

00

0

201

10

00

01

11

1

323

33

35

68

810

14

4426

2627

2731

4654

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81

5612

912

914

214

218

224

828

928

933

241

0

6857

357

363

663

684

211

2412

9912

9914

8618

42

8022

7622

7625

4425

4433

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1251

7851

7859

0672

78

9282

2682

2692

2392

2312

251

1636

618

749

1874

921

367

2630

5

104

2750

527

505

3090

230

902

4114

254

888

6281

262

812

7156

288

062

116

8634

986

349

9704

497

044

1292

6817

2448

1972

1419

7214

2246

8427

6276

128

2564

4725

6447

2883

5128

8351

3842

7351

2502

5859

3858

5938

6674

9082

0618

140

7262

0172

6201

8167

4481

6744

1088

680

1451

756

1659

543

1659

543

1890

431

2323

911

152

1972

157

1972

157

2218

282

2218

282

2957

154

3943

264

4507

162

4507

162

5134

124

6310

848

164

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351

5160

351

5804

744

5804

744

7738

814

1031

8996

1179

4091

1179

4091

1343

4478

1651

3050

176

1306

0772

1306

0772

1469

2426

1469

2426

1958

8686

2611

9004

2985

1826

2985

1826

3400

3540

4179

4712

188

3208

0056

3208

0056

3608

8522

3608

8522

4811

5930

6415

6020

7332

3427

7332

3427

8352

0644

1026

5596

2

200

7667

7496

7667

7496

8625

9890

8625

9890

1150

1006

615

3348

868

1752

5931

617

5259

316

1996

3235

424

5367

882

212

1787

7028

317

8770

283

2011

1315

220

1113

152

2681

4642

435

7531

400

4086

1270

640

8612

706

4654

3695

457

2065

334

224

4073

8219

040

7382

190

4582

9975

845

8299

758

6110

5941

081

4750

620

9311

5130

993

1151

309

1060

6418

7413

0362

2708

236

9089

8565

190

8985

651

1022

6012

1310

2260

1213

1363

4580

0918

1795

0926

2077

6697

2020

7766

9720

2366

5990

9129

0875

3703

248

1989

0020

7219

8900

2072

2237

6164

5022

3761

6450

2983

4740

9039

7797

4852

4546

2740

8845

4627

4088

5178

4952

5663

6480

7716

260

4273

9785

1642

7397

8516

4808

2095

6848

0820

9568

6410

9244

4485

4791

3920

9769

0687

9697

6906

8796

1112

7587

228

1367

6730

320

272

9029

7704

1690

2977

0416

1015

8468

546

1015

8468

546

1354

4593

866

1805

9479

228

2063

9439

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2063

9439

888

2350

9622

436

2889

5264

904

284

1877

7609

455

1877

7609

455

2112

4777

990

2112

4777

990

2816

6327

222

3755

5131

344

4292

0200

803

4292

0200

803

4888

8805

889

6008

8352

145


Tabl

e 35

The

tab

le s

how

s th

e de

com

posi

tion

of

the

Four

ier

coeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

fun

ctio

n hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f M

23 -

part

I

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

χ9

−1

−1

00

00

00

00

230

00

00

00

00

470

00

01

10

00

710

01

13

32

24

951

44

424

2321

2124

119

010

2323

110

115

113

113

122

143

357

104

104

555

552

547

547

604

167

1121

745

045

022

6622

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8325

00

191

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46

215

132

2945

6066

6066

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076

3106

231

062

3400

2

239

455

9865

2006

120

061

1027

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311

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118

335

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2082

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639

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1938

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7219

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6

455

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7200

4775

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614

7289

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7527

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6074

520

7560

7419

175

6074

191

8280

7894

0

479

7632

762

1679

0846

434

3437

978

3434

3797

817

5537

6093

1762

9976

1017

6299

7173

1762

9971

7319

3090

5618

503

1743

6751

3836

2443

878

4701

896

7847

0189

640

1065

9821

4028

1134

0140

2811

2844

4028

1128

4444

1173

7282

527

3909

6804

8601

0367

917

5927

8672

1759

2786

7289

9192

7075

9030

9991

1790

3099

8389

9030

9983

8998

9110

2068

551

8611

3417

1894

5295

1338

7520

7902

3875

2079

0219

8065

3365

319

8926

8336

819

8926

8244

619

8926

8244

621

7872

1203

2

575

1865

7820

841

0466

4777

8395

8534

5083

9585

3450

4291

2264

213

4309

8789

349

4309

8788

160

4309

8788

160

4720

3452

886


Tabl

e 36

The

tab

le s

how

s th

e de

com

posi

tion

of

the

Four

ier

coeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

fun

ctio

n hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f M

23 -

part

II

−D

χ10

χ11

χ12

χ13

χ14

χ15

χ16

χ17

−1

00

00

00

00

231

00

00

00

0

471

11

11

12

3

7111

1111

1114

1412

26

9568

6880

8087

8796

182

119

389

389

444

444

496

496

508

1003

143

1811

1811

2111

2111

2324

2324

2454

4777

167

7661

7661

8892

8892

9848

9848

1024

620

084

191

2924

529

245

3405

234

052

3759

637

596

3939

476

950

215

1036

9510

3695

1205

8612

0586

1333

0613

3306

1392

0427

2375

239

3435

2534

3525

3998

1939

9819

4416

5844

1658

4620

2890

3231

263

1076

965

1076

965

1252

987

1252

987

1384

656

1384

656

1447

084

2830

347

287

3212

975

3212

975

3738

996

3738

996

4130

944

4130

944

4319

566

8446

311

311

9184

195

9184

195

1068

6484

1068

6484

1180

8192

1180

8192

1234

3520

2413

9819

335

2526

0382

2526

0382

2939

4631

2939

4631

3247

7570

3247

7570

3395

6132

6640

0838

359

6713

6828

6713

6828

7812

1437

7812

1437

8631

8725

8631

8725

9023

8598

1764

7019

4

383

1729

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817

2966

448

2012

7197

420

1271

974

2223

8534

522

2385

345

2324

9959

645

4660

159

407

4332

2552

043

3225

520

5041

1350

550

4113

505

5570

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555

7004

065

5823

1336

011

3875

4982

431

1057

3869

7910

5738

6979

1230

4186

2712

3041

8627

1359

4973

4813

5949

7348

1421

3066

9427

7943

0485

455

2520

2652

2225

2026

5222

2932

6643

4629

3266

4346

3240

3408

1432

4034

0814

3387

6076

7866

2467

5836

479

5876

6317

6558

7663

1765

6838

2733

9768

3827

3397

7555

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3175

5566

9131

7899

1407

6815

4471

7714

8

503

1342

7082

844

1342

7082

844

1562

4224

127

1562

4224

127

1726

3391

744

1726

3391

744

1804

8043

408

3529

3998

406

527

3010

3271

271

3010

3271

271

3502

9285

569

3502

9285

569

3870

4205

367

3870

4205

367

4046

3558

496

7912

8667

059

551

6630

9028

068

6630

9028

068

7715

9558

254

7715

9558

254

8525

4464

108

8525

4464

108

8912

9562

686

1742

9791

3365

575

1436

6250

5416

1436

6250

5417

1671

7096

8143

1671

7096

8143

1847

0893

4751

1847

0893

4751

1931

0494

7248

3776

2730

3791


Tabl

e 37

The

tab

le s

how

s th

e de

com

posi

tion

of

the

Four

ier

coeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

fun

ctio

n hg,3 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f M

23 -

part

I

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

χ9

−9

10

00

00

00

0

150

00

00

10

00

390

10

01

00

00

630

00

02

12

22

871

22

210

1210

1012

111

08

1212

7271

6565

76

135

129

6666

317

324

327

327

354

159

814

827

827

814

5614

5114

3914

3915

84

183

2553

611

0811

0856

1456

6656

5056

5061

86

207

9019

6939

9639

9620

527

2057

120

585

2058

522

554

231

301

6612

1357

413

574

6920

069

575

6953

969

539

7616

0

255

979

2122

943

224

4322

422

1297

2221

2822

2072

2220

7224

3296

279

2892

6397

713

1140

1311

4066

9671

6728

2667

2862

6728

6273

6834

303

8477

1859

2537

9810

3798

1019

4229

319

5036

119

5025

319

5025

321

3617

8

327

2348

551

6995

1058

004

1058

004

5406

043

5430

193

5430

053

5430

053

5947

048

351

6322

913

9015

828

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028

4255

014

5311

0614

5932

2014

5933

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5933

1915

9834

98

375

1643

1436

1615

973

9813

073

9813

037

8087

3937

9748

2337

9745

6637

9745

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10

399

4157

3391

4280

518

6982

6218

6982

6295

5754

8795

9885

0295

9881

4895

9881

4810

5130

940

423

1022

444

2249

8248

4602

3364

4602

3364

2352

2048

623

6247

135

2362

4736

423

6247

364

2587

4561

2

447

2456

955

5404

5257

1105

4036

411

0540

364

5649

9952

356

7449

989

5674

4937

556

7449

375

6214

9463

2

471

5768

329

1269

1180

725

9601

046

2596

0104

613

2682

7403

1332

6055

7513

3260

4789

1332

6047

8914

5951

6642

495

1326

9156

2919

0777

659

7070

540

5970

7054

030

5172

7864

3064

9823

5530

6498

2875

3064

9828

7533

5689

0620

519

2993

0003

6584

7866

813

4690

7692

1346

9076

9268

8414

4714

6914

0964

1469

1409

5094

6914

0950

9475

7257

3738

543

6631

2927

1458

8488

2029

8397

6756

2983

9767

5615

2515

1250

015

3177

9333

415

3177

9161

615

3177

9161

616

7766

4043

6

567

1444

4101

231

7775

1973

6499

9930

0464

9999

3004

3322

2076

673

3336

6564

639

3336

6565

723

3336

6565

723

3654

4317

696


Tabl

e 38

The

tab

le s

how

s th

e de

com

posi

tion

of

the

Four

ier

coeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

fun

ctio

n hg,3 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f M

23 -

part

II

−D

χ10

χ11

χ12

χ13

χ14

χ15

χ16

χ17

−9

00

00

00

00

150

00

00

00

0

391

10

01

10

1

634

46

66

68

13

8741

4142

4249

4948

97

111

220

220

254

254

280

280

302

582

135

1099

1099

1278

1278

1418

1418

1462

2879

159

4783

4783

5568

5568

6143

6143

6460

1259

4

183

1892

018

920

2196

721

967

2430

824

308

2534

249

625

207

6848

268

482

7974

979

749

8805

588

055

9219

418

0159

231

2320

6423

2064

2699

1226

9912

2983

4729

8347

3116

6260

9707

255

7399

8473

9984

8611

6986

1169

9513

7495

1374

9950

3819

4543

8

279

2243

393

2243

393

2610

275

2610

275

2884

379

2884

379

3014

774

5896

262

303

6500

043

6500

043

7563

999

7563

999

8357

121

8357

121

8738

192

1708

6836

327

1810

2016

1810

2016

2106

3317

2106

3317

2327

3934

2327

3934

2432

9868

4758

0312

351

4864

1673

4864

1673

5660

2448

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9370

6253

9370

6538

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6136

7

375

1265

8639

112

6586

391

1472

9844

314

7298

443

1627

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316

2753

713

1701

4660

833

2736

007

399

3199

5466

631

9954

666

3723

1329

437

2313

294

4113

7002

641

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026

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484

1030

812

423

7875

0073

178

7500

731

9163

6044

891

6360

448

1012

5011

4810

1250

1148

1058

5117

6220

6999

0475

447

1891

4838

1718

9148

3817

2201

0053

0222

0100

5302

2431

9073

5624

3190

7356

2542

4671

4649

7191

7548

471

4442

0401

7344

4204

0173

5168

9085

8051

6890

8580

5711

1938

7557

1119

3875

5970

7655

6211

6761

9110

8

495

1021

6574

441

1021

6574

441

1188

8393

029

1188

8393

029

1313

5596

047

1313

5596

047

1373

2710

656

2685

5037

539

519

2304

7036

703

2304

7036

703

2681

8346

389

2681

8346

389

2963

1903

464

2963

1903

464

3097

8745

714

6058

0719

175

543

5105

9233

415

5105

9233

415

5941

4411

523

5941

4411

523

6564

7584

650

6564

7584

650

6863

1657

584

1342

1292

9307

567

1112

2199

3159

1112

2199

3159

1294

2190

9124

1294

2190

9124

1429

9970

6146

1429

9970

6146

1494

9955

8772

2923

5482

3899


Tabl

e 39

The

tabl

e sh

ows t

he d

ecom

posi

tion

of th

e Fo

urie

r coe

ffici

ents

mul

tiply

ing q−D/24 in

the

func

tion hg,1 2

(τ) i

nto

irred

ucib

le re

pres

enta

tions

χn o

f McL

- pa

rt I

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

−1

−1

00

00

00

0

230

00

01

00

0

470

00

00

00

0

710

00

01

10

0

950

11

12

11

1

119

01

21

77

55

143

03

88

2322

2424

167

15

2729

9392

101

101

191

013

103

113

336

336

386

386

215

237

357

381

1192

1191

1370

1370

239

612

211

8112

9239

1139

0945

4145

41

263

1835

736

7339

9012

260

1225

914

231

1423

1

287

4910

6510

971

1198

736

504

3650

342

467

4246

7

311

135

2983

3129

634

082

1043

8110

4379

1213

7512

1375

335

379

8247

8612

794

007

2869

1928

6918

3338

6533

3865

359

995

2179

622

8754

2494

3276

2672

7626

7088

7304

8873

04

383

2564

5623

158

9479

6432

1819

6454

919

6454

522

8605

022

8605

0

407

6386

1405

6814

7613

816

1001

649

2087

049

2086

957

2573

157

2573

1

431

1562

234

3337

3603

184

3931

104

1200

9753

1200

9751

1397

5118

1397

5118

455

3716

781

7799

8587

440

9367

446

2862

5805

2862

5800

3330

9272

3330

9272

479

8675

619

0749

820

0245

8321

8458

8266

7466

5466

7466

5277

6692

7477

6692

74

503

1980

1443

5704

445

7511

9249

9088

3615

2506

325

1525

0632

117

7460

315

1774

6031

5

527

4441

7397

6982

810

2575

149

1119

0211

534

1912

961

3419

1295

839

7863

480

3978

6348

0

551

9779

9721

5176

6322

5940

987

2464

7785

475

3141

922

7531

4192

087

6380

147

8763

8014

7

575

2119

404

4662

2397

4895

1878

353

4024

793

1631

7204

5816

3172

0453

1898

7319

7218

9873

1972


Tabl

e 40

The

tab

le s

how

s th

e de

com

posi

tion

of

the

Four

ier

coeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

fun

ctio

n hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f McL

- pa

rt II

−D

χ9

χ10

χ11

χ12

χ13

χ14

χ15

χ16

−1

00

00

00

00

230

00

00

00

0

470

10

00

00

0

710

11

10

11

2

953

53

54

55

8

119

821

2225

2727

3246

143

4999

9212

012

313

814

621

4

167

193

396

402

507

536

572

628

909

191

765

1535

1513

1946

2039

2212

2385

3459

215

2657

5375

5400

6874

7271

7783

8486

1226

9

239

8900

1788

117

816

2281

624

056

2588

728

072

4062

6

263

2774

555

882

5596

371

465

7550

081

007

8809

512

7410

287

8303

316

6945

1667

6621

3315

2251

5324

1943

2626

9438

0029

311

2369

1947

6743

4769

8260

9553

6438

0369

1150

7511

3210

8640

5

335

6523

1413

1187

813

1141

216

7684

517

7048

319

0166

620

6559

229

8787

9

359

1732

646

3485

567

3486

207

4456

239

4706

100

5053

141

5490

489

7941

451

383

4465

544

8981

556

8980

403

1148

1470

1212

3772

1302

0310

1414

4455

2045

9305

407

1118

2141

2249

3230

2249

4840

2875

6211

3036

7390

3260

8993

3542

8696

5124

4665

431

2729

6592

5490

3746

5490

1023

7018

7986

7411

6978

7959

3979

8646

9919

1250

7338

7

455

6505

5014

1308

5609

013

0859

914

1672

8929

717

6659

468

1897

0476

020

6102

830

2981

1203

9

479

1517

0109

030

5132

577

3051

2639

339

0081

042

4119

2213

244

2353

736

4805

7598

769

5120

690

503

3465

9729

569

7161

444

6971

7013

889

1261

397

9411

7661

110

1068

7552

1098

0395

9815

8823

3526

527

7770

8436

715

6304

2221

1563

0286

6419

9820

0428

2110

0921

6522

6596

3373

2461

7744

7435

6078

4919

551

1711

6694

1634

4291

2979

3442

9319

8844

0145

9954

4647

9516

8549

9124

9366

5422

6106

8078

4341

3933

575

3708

4763

9174

5931

9124

7459

2903

0995

3604

6829

1007

0049

592

1081

3885

859

1174

8391

635

1699

3217

734


Tabl

e 41

The

tab

le s

how

s th

e de

com

posi

tion

of

the

Four

ier

coeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

fun

ctio

n hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f McL

- pa

rt II

I

−D

χ17

χ18

χ19

χ20

χ21

χ22

χ23

χ24

−1

00

00

00

00

230

00

00

00

0

470

00

00

00

1

712

11

11

11

2

958

88

1010

1011

11

119

4646

4653

5355

5960

143

214

221

221

261

261

263

279

279

167

909

928

928

1076

1076

1110

1175

1175

191

3459

3561

3561

4167

4167

4252

4487

4489

215

1226

912

607

1260

714

687

1468

715

064

1589

915

900

239

4062

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815

4181

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824

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950

5268

352

683

263

1274

1013

1025

1310

2515

2796

1527

9615

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5131

287

3800

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6320

4563

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7137

4926

7749

2679

311

1086

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568

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1303

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9010

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1084

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7382

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575

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319

1748

2746

319

2039

6558

046

2039

6558

046

2088

6051

640

2202

8253

515

2202

8253

520


Tabl

e 42

The

tabl

e sh

ows t

he d

ecom

posi

tion

of th

e Fo

urie

r coe

ffici

ents

mul

tiply

ing q−D/24 in

the

func

tion hg,3 2

(τ) i

nto

irred

ucib

le re

pres

enta

tions

χn o

f McL

- pa

rt I

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

−9

10

00

00

00

150

01

00

00

0

390

00

10

00

0

630

01

00

00

0

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01

11

10

0

111

00

41

33

22

135

10

65

1212

1414

159

13

2423

5555

6262

183

25

7272

219

219

247

247

207

222

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777

777

905

905

231

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865

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255

1825

125

5627

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3425

592

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4431

1469

9771

3814

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7138


Tabl

e 43

The

tab

le s

how

s th

e de

com

posi

tion

of

the

Four

ier

coeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

fun

ctio

n hg,3 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f McL

- pa

rt II

−D

χ9

χ10

χ11

χ12

χ13

χ14

χ15

χ16

−9

00

00

00

00

150

00

00

00

0

390

00

00

01

0

630

00

10

11

0

871

12

22

26

4

111

713

1015

1419

2125

135

2454

5675

7881

9513

0

159

129

254

244

320

334

365

398

562

183

482

975

986

1251

1328

1414

1566

2235

207

1776

3566

3536

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4792

5173

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8093

231

5969

1202

612

064

1539

416

274

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519

023

2745

1

255

1915

438

471

3838

049

139

5184

655

752

6053

387

506

279

5780

911

6393

1165

1014

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8216

8786

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303

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6895

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9

351

1255

917

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910

2525

280

3228

953

3409

380

3661

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3977

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3267

135

6572

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301

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5430

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9222

2873

8269

4796

7796

1384

5883

7196

7563

9095

4962

6113

1559

7652

6


Tabl

e 44

The

tab

le s

how

s th

e de

com

posi

tion

of

the

Four

ier

coeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

fun

ctio

n hg,3 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f McL

- pa

rt II

I

−D

χ17

χ18

χ19

χ20

χ21

χ22

χ23

χ24

−9

00

00

00

00

150

00

00

00

0

390

00

00

00

0

630

00

11

11

1

874

55

55

56

6

111

2527

2732

3231

3434

135

130

132

132

153

153

161

168

168

159

562

583

583

685

685

695

733

733

183

2235

2298

2298

2671

2671

2743

2898

2898

207

8093

8336

8336

9753

9753

9968

1050

310

503

231

2745

128

225

2822

532

895

3289

533

718

3557

635

576

255

8750

690

070

9007

010

5135

1051

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7589

1134

6711

3467

279

2653

6927

2959

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5931

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3183

7532

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3439

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303

7688

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1067

7910

6792

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519

2726

1367

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7750

3272

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2335

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7823

543

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6213

5621

5572

4916

8236

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1682

3674

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3737

7829

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2908

3804

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526

1353

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287

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287

1579

0750

779

1579

0750

779

1616

9749

982

1705

4036

789

1705

4036

789


Tabl

e 45

The

tabl

e sh

ows

the

deco

mpo

siti

on o

f the

Fou

rier

coe

ffici

ents

mul

tipl

ying

q−D/24 in

the

func

tion

hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χ

n o

f HS

- par

t I −

Dχ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

−1

−1

00

00

00

0

230

00

00

00

0

470

00

00

00

0

710

00

10

01

0

950

23

25

53

6

119

02

819

1414

2124

143

016

4480

9090

9213

1

167

450

174

357

339

339

407

515

191

720

368

113

3413

7213

7215

2120

39

215

3367

223

7247

8747

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3571

12

239

9922

9279

2915

766

1588

715

887

1792

923

777

263

330

7047

2473

349

595

4939

249

392

5634

874

177

287

948

2120

773

979

1476

9514

8048

1480

4816

7874

2219

11

311

2759

6024

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1115

4226

2842

2054

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6333

38

335

7516

1662

3358

1198

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707

1162

651

1162

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1320

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556

359

2010

844

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1543

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3088

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208

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4631

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383

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111

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379

5577

979

5817

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5817

190

4089

211

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73

407

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399

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019

9293

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2429

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00

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3156

5969

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124

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5925

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617

3892

768

479

1754

960

3862

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1351

6536

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289

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6456

288

527

8990

981

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2806

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551

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7924

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2509

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3050

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1634

6616

3569

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28

575

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633

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6608

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1666

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9893

6608

5198

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0961

6830

9912

7519

19


Tabl

e 46

The

tabl

e sh

ows

the

deco

mpo

siti

on o

f the

Fou

rier

coe

ffici

ents

mul

tipl

ying

q−D/24 in

the

func

tion

hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χ

n o

f HS

- par

t II −

Dχ9

χ10

χ11

χ12

χ13

χ14

χ15

χ16

−1

00

00

00

00

230

01

00

00

0

470

10

01

00

0

713

23

43

33

2

9514

1915

1518

1717

22

119

7985

9494

9510

610

611

7

143

375

432

413

412

449

478

478

575

167

1585

1739

1776

1777

1887

2061

2061

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191

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6780

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9247

215

2145

423

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239

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078

950

8468

691

864

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263

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7836

2653

6528

8353

2883

5333

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287

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2073

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8797

7918

5785

9693

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9310

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5

311

1901

066

2111

639

2112

736

2112

736

2263

145

2458

353

2458

353

2896

427

335

5228

981

5811

159

5809

370

5809

369

6225

096

6760

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6760

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7968

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359

1389

7177

1543

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1796

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5541

383

3580

4287

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5466

3978

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3978

0943

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4415

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4629

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8967

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1067

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5949

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6648

460

431

2188

7965

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3206

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2605

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2991

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2829

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933

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455

5216

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057

9649

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5796

6627

057

9666

271

6210

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967

4519

766

6745

1976

679

4956

158

479

1216

4640

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2677

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7768

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1572

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503

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4041

5030

8820

3718

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3308

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3535

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527

6231

3799

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6923

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4869

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7418

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7280

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5515

551

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256

1525

1101

926

1525

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928

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1774

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1774

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2091

5725

781

575

2973

8144

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3304

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3304

2340

266

3304

2340

266

3540

2559

005

3844

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233

3844

9273

233

4531

5309

735


Tabl

e 47

The

tabl

e sh

ows

the

deco

mpo

siti

on o

f the

Fou

rier

coe

ffici

ents

mul

tipl

ying

q−D/24 in

the

func

tion

hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χ

n o

f HS

- par

t III −

Dχ17

χ18

χ19

χ20

χ21

χ22

χ23

χ24

−1

00

00

00

00

230

00

00

00

0

470

00

11

11

1

714

56

46

79

9

9526

2735

4240

5356

67

119

162

164

202

215

221

286

317

361

143

740

756

941

1054

1046

1371

1487

1743

167

3180

3230

4009

4374

4396

5744

6289

7290

191

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Tabl

e 48

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mpo

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on o

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rier

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the

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Tabl

e 49

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e sh

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mpo

siti

on o

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rier

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ffici

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mul

tipl

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the

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n o

f HS

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Tabl

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e sh

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mpo

siti

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tipl

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n o

f HS

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Tabl

e 51

The

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how

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e de

com

posi

tion

of t

he F

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ts m

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the

func

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f U6(2) -

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rt I

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Tabl

e 52

The

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s th

e de

com

posi

tion

of t

he F

ouri

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oeffi

cien

ts m

ulti

plyi

ng q

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the

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tion

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nto

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tati

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f U6(2) -

pa

rt II

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00

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Tabl

e 53

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

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oeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

func

tion

hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U6(2) -

pa

rt II

I

−D

χ20

χ21

χ22

χ23

χ24

χ25

χ26

χ27

χ28

−1

00

00

00

00

0

230

00

00

00

00

470

00

00

00

00

711

10

10

00

00

950

21

21

11

11

119

48

57

54

44

4

143

1224

2224

2022

2424

24

167

5989

8185

8386

9797

97

191

204

281

291

296

302

345

388

388

388

215

750

942

982

991

1064

1194

1360

1360

1360

239

2420

2932

3180

3191

3473

4006

4564

4564

4564

263

7688

9022

9799

9816

1085

612

481

1426

614

266

1426

6

287

2275

426

332

2898

329

006

3223

237

369

4272

842

728

4272

8

311

6531

374

730

8232

382

356

9203

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6605

1220

2712

2027

1220

27

335

1791

6220

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3554

3360

8033

6080

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359

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8929

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2976

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76

383

1227

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781

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529

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407

3076

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5031

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5764

588

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588

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7505

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281

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420

1055

9896

1228

3559

1407

2625

1407

2625

1407

2625

455

1789

2382

2015

9114

2238

3595

2238

3781

2516

7016

2927

4597

3354

0710

3354

0710

3354

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479

4171

4567

4697

9165

5217

9270

5217

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5867

5953

6826

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7821

5160

7821

5160

7821

5160

503

9531

9311

1073

1205

911

9196

613

1191

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813

4058

491

1559

6844

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8705

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1787

0575

517

8705

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527

2136

9102

224

0528

777

2672

0376

926

7204

168

3005

3968

934

9688

430

4006

6980

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0669

808

4006

6980

8

551

4707

2001

952

9752

466

5885

1885

258

8519

370

6619

9003

977

0250

507

8825

5852

888

2558

528

8825

5852

8

575

1019

8155

8311

4759

8073

1274

9856

2912

7498

6283

1434

2030

4416

6881

5367

1912

1486

0119

1214

8601

1912

1486

01


Tabl

e 54

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

func

tion

hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U6(2) -

pa

rt IV

−D

χ29

χ30

χ31

χ32

χ33

χ34

χ35

χ36

χ37

−1

00

00

00

00

0

230

00

00

00

00

471

00

00

00

00

711

00

00

01

00

954

31

11

13

01

119

119

44

46

124

11

143

4339

2626

2630

4730

44

167

140

135

105

105

105

124

181

141

208

191

504

495

430

430

430

476

641

581

769

215

1659

1647

1512

1512

1512

1679

2189

2112

2800

239

5391

5373

5106

5106

5106

5576

7089

7143

9196

263

1651

316

488

1598

115

981

1598

117

469

2195

622

579

2903

4

287

4894

448

909

4796

447

964

4796

452

139

6497

067

811

8640

1

311

1388

1213

8765

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5213

7052

1370

5214

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1848

7619

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2475

40

335

3808

4638

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3777

4537

7745

3777

4540

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5069

6453

5979

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73

359

1009

253

1009

164

1003

898

1003

898

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898

1089

307

1345

063

1426

142

1809

476

383

2597

544

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424

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431

2588

431

2588

431

2806

561

3461

147

3677

682

4660

418

407

6498

438

6498

279

6483

169

6483

169

6483

169

7029

317

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9215

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1167

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1585

3998

1585

3785

1582

8769

1582

8769

1582

8769

1715

7026

2113

2786

2250

2085

2849

5813

455

3776

9072

3776

8797

3772

7937

3772

7937

3772

7937

4089

2977

5035

3789

5364

4469

6792

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0558

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4244

9535

3051

1173

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5107

300

1583

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7

503

2011

3595

220

1135

486

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3024

220

1030

242

2010

3024

221

7864

013

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6799

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5875

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9

527

4509

0047

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5301

3

551

9931

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599

2844

595

9928

4459

510

7591

2674

1324

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1196

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1787

2234

81

575

2151

5222

6521

5152

1277

2151

1188

7421

5111

8874

2151

1188

7423

3103

3239

2868

6125

6030

5921

4800

3872

0973

31


Tabl

e 55

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

func

tion

hg,1 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U6(2) -

pa

rt V

−D

χ38

χ39

χ40

χ41

χ42

χ43

χ44

χ45

χ46

−1

00

00

00

00

0

230

00

00

00

00

470

00

00

00

00

710

00

01

11

00

952

22

33

33

32

119

911

1112

1717

1821

17

143

4955

5567

6969

8794

91

167

199

228

228

265

295

295

362

409

415

191

786

881

881

1049

1088

1088

1384

1557

1631

215

2771

3105

3105

3657

3864

3864

4892

5575

5869

239

9251

1031

110

311

1223

012

681

1268

116

220

1845

219

609

263

2894

332

259

3225

938

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339

773

5083

157

985

6176

0

287

8656

096

303

9630

311

4151

1182

9611

8296

1516

7517

2971

1847

63

311

2472

8627

5091

2750

9132

5787

3381

6233

8162

4335

0349

4841

5289

70

335

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0275

6753

7567

5389

6888

9291

0992

9109

1192

384

1360

994

1456

375

359

1808

802

2010

778

2010

778

2382

438

2469

363

2469

363

3168

960

3618

306

3873

101

383

4661

495

5180

729

5180

729

6140

008

6359

343

6359

343

8164

490

9321

948

9982

360

407

1167

4769

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1297

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1537

5595

1592

7891

1592

7891

2044

9076

2335

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8800

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2849

8396

3166

8560

3166

8560

3753

2560

3886

9434

3886

9434

4991

1204

5699

4271

6105

0043

455

6792

3794

7547

8392

7547

8392

8945

0970

9264

3752

9264

3752

1189

6173

213

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564

1455

2748

6

479

1583

8939

617

5998

083

1759

9808

320

8588

414

2160

0731

421

6007

314

2773

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931

6772

980

3393

5918

6

503

3618

8887

140

2117

862

4021

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4935

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6911

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1420

9411

3716

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551

1787

2044

1119

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8294

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3722

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9293

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52

575

3872

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3193

4302

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9350

9915

2355

5280

3105

0452

8031

0504

6781

1858

1077

4422

2864

8297

1325

08


Tabl

e 56

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

func

tion

hg,3 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U6(2) -

pa

rt I

−D

χ1

χ2

χ3

χ4

χ5

χ6

χ7

χ8

χ9

χ10

−9

10

00

00

00

00

150

01

00

00

00

0

390

00

10

00

00

0

630

01

00

10

00

0

871

01

10

10

01

1

111

00

41

02

00

22

135

10

54

15

20

55

159

12

148

413

67

1414

183

22

2419

1030

2420

4343

207

28

6240

4279

6666

113

113

231

516

140

124

128

213

228

223

342

342

255

1054

379

334

423

585

648

693

964

964

279

1710

794

692

812

4216

3919

5220

3727

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82

303

3029

925

7425

5636

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0254

3458

8976

9776

97

327

6471

666

5569

4710

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1214

915

072

1626

721

026

2102

6

351

122

1817

1736

618

206

2708

731

983

3984

843

501

5545

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450

375

287

4447

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547

011

7019

982

214

1035

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2936

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2014

3220

399

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310

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1177

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7721

2060

6325

9946

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9735

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6768

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6220

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616

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924

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473

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2399

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2561

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1835

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5825

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7823

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8212

3491

729

1234

9172

9


Tabl

e 57

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

func

tion

hg,3 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U6(2) -

pa

rt II

−D

χ11

χ12

χ13

χ14

χ15

χ16

χ17

χ18

χ19

−9

00

00

00

00

0

150

00

00

00

00

390

00

00

00

00

630

00

00

00

00

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00

00

00

01

111

00

00

02

24

4

135

11

10

08

810

9

159

55

511

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3040

44

183

2626

2630

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010

012

713

9

207

9797

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035

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7

231

363

363

363

436

475

1106

1106

1310

1596

255

1172

1172

1172

1484

1607

3458

3458

4034

5088

279

3638

3638

3638

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4842

1024

810

248

1185

615

102

303

1061

110

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112

975

1429

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3

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3974

181

284

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8028

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286

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9


Tabl

e 58

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

ulti

plyi

ng q

−D/24 in

the

func

tion

hg,3 2

(τ) i

nto

irre

duci

ble

repr

esen

tati

ons χn o

f U6(2) -

pa

rt II

I

−D

χ20

χ21

χ22

χ23

χ24

χ25

χ26

χ27

χ28

−9

00

00

00

00

0

150

00

00

00

00

390

10

00

00

00

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10

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204

193

193

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4863

288

8501

021

9870

9618

698

7096

186

1110

3569

7812

9196

4951

1480

3495

6814

8034

9568

1480

3495

68


Tabl

e 59

The

tab

le s

how

s th

e de

com

posi

tion

of t

he F

ouri

er c

oeffi

cien

ts m

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ng q

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the

func

tion

hg,3 2

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nto

irre

duci

ble

repr

esen

tati

ons χn o

f U6(2) -

pa

rt IV

−D

χ29

χ30

χ31

χ32

χ33

χ34

χ35

χ36

χ37

−9

00

00

00

00

0

150

00

00

00

00

390

00

00

00

00

631

10

00

01

00

872

20

00

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02

111

88

33

34

83

5

135

2424

1313

1317

3516

32

159

9595

6969

6977

118

8912

3

183

322

322

267

267

267

306

424

367

517

207

1124

1124

1011

1011

1011

1113

1471

1390

1822

231

3633

3633

3411

3411

3411

3763

4830

4796

6268

255

1144

411

444

1101

911

019

1101

912

011

1514

015

497

1986

4

279

3416

434

164

3337

233

372

3337

236

393

4550

647

238

6049

1

303

9854

198

541

9710

497

104

9710

410

5478

1310

1413

7526

1749

34

327

2729

1827

2918

2703

7227

0372

2703

7229

3697

3636

5638

3784

4879

25

351

7320

5573

2055

7276

1272

7612

7276

1278

9280

9750

5310

3303

313

1027

4

375

1901

107

1901

107

1893

498

1893

498

1893

498

2053

906

2534

016

2690

675

3411

830

399

4801

562

4801

562

4788

741

4788

741

4788

741

5191

571

6399

104

6805

582

8621

691

423

1180

7813

1180

7813

1178

6521

1178

6521

1178

6521

1277

7678

1574

1552

1675

6387

2122

5130

447

2835

1578

2835

1578

2831

6673

2831

6673

2831

6673

3069

0911

3779

4342

4025

8951

5097

5995

471

6655

7162

6655

7162

6650

0652

6650

0652

6650

0652

7207

5158

8873

6216

9456

0774

1197

2505

6

495

1530

5616

415

3056

164

1529

6572

215

2965

722

1529

6572

216

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974

2040

5573

321

7516

112

2753

5524

4

519

3452

1296

934

5212

969

3450

6977

734

5069

777

3450

6977

737

3954

834

4602

6634

249

0719

721

6211

8391

7

543

7647

4065

076

4740

650

7645

1615

176

4516

151

7645

1615

182

8475

182

1019

6072

6410

8722

5658

1376

1741

13

567

1665

6967

1316

6569

6713

1665

3480

0516

6534

8005

1665

3480

0518

0465

8930

2220

8849

0723

6838

0807

2997

7639

88


Tabl

e 60

The

tab

le s

how

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tion

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150

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00

00

00

390

00

00

00

00

630

00

10

00

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871

11

12

22

20

111

67

79

88

119

9

135

2832

3238

4545

5158

54

159

130

145

145

175

179

179

225

250

255

183

503

566

566

662

715

715

891

1012

1044

207

1846

2060

2060

2457

2536

2536

3233

3662

3861

231

6220

6951

6951

8205

8608

8608

1095

112

474

1321

5

255

1993

722

198

2219

826

322

2727

827

278

3493

439

779

4238

1

279

6035

567

201

6720

179

550

8272

882

728

1058

7812

0825

1288

99

303

1751

4719

4787

1947

8723

0873

2392

0023

9200

3068

3135

0044

3742

14

327

4875

5554

2191

5421

9164

2249

6661

8866

6188

8544

2197

5422

1043

352

351

1310

841

1457

089

1457

089

1726

958

1788

772

1788

772

2296

034

2621

064

2805

657

375

3410

897

3791

267

3791

267

4492

367

4655

102

4655

102

5975

070

6822

630

7304

754

399

8623

112

9583

033

9583

033

1135

7471

1176

2638

1176

2638

1510

2657

1724

4669

1846

8738

423

2122

2871

2358

4843

2358

4843

2794

9916

2895

0941

2895

0941

3717

1558

4244

7872

4546

5426

447

5097

9357

5664

8802

5664

8802

6713

8492

6952

8362

6952

8362

8928

2411

1019

5512

910

9216

491

471

1197

1986

513

3032

283

1330

3228

315

7661

257

1632

8196

016

3281

960

2096

7356

223

9444

613

2565

0971

1

495

2753

6288

830

5972

205

3059

7220

536

2631

248

3755

2463

937

5524

639

4822

4514

055

0717

850

5899

9952

1

519

6211

7238

769

0217

270

6902

1727

081

8019

198

8471

2195

884

7121

958

1087

8694

4612

4235

5809

1330

9995

39

543

1376

1908

4015

2913

4899

1529

1348

9918

1229

9925

1876

7000

2918

7670

0029

2410

1084

3027

5236

4303

2948

8236

57

567

2997

7392

8133

3088

3649

3330

8836

4939

4767

6846

4087

9908

5040

8799

0850

5249

9185

6059

9550

2658

6423

5200

20


Received: 1 July 2014 Accepted: 15 June 2015

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