Effective bounds for Fourier coefficients of certain weakly holomorphic modular forms

Journal of Number Theory 191 (2018) 384–395

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Journal of Number Theory www.elsevier.com/locate/jnt

Effective bounds for Fourier coefficients of certain weakly holomorphic modular forms Daniel Garbin Department of Mathematics and Computer Science, Queensborough Community College, 222-05 56th Avenue, Bayside, NY 11364, USA

a r t i c l e

i n f o

Article history: Received 21 January 2018 Received in revised form 4 March 2018 Accepted 11 March 2018 Available online 17 April 2018 Communicated by L. Smajlovic Keywords: Modular forms Automorphic forms Moonshine type arithmetic groups Fourier coefficients Holomorphic Eisenstein series Partition function

a b s t r a c t In Jorgenson et al. (2016) [JST 16a], the authors derived generators for the function fields associated to certain low genus arithmetic surfaces realized through the action of the discrete Fuchsian group Γ0 (N )+ /{±1} on the upper half plane. In particular, they construct modular forms which are analogs to the modular discriminant and the Klein j-invariant of the full modular group PSL(2, Z). In this article, we produce effective and practical bounds for the Fourier coefficients in the q-expansion of such generators, thus allowing for rigorous numerical inspection of the generators. © 2018 Elsevier Inc. All rights reserved.

1. Introduction The action of the full modular group PSL(2, Z) on the upper-half plane H yields the quotient space X1 ∼ = PSL(2, Z)\H which is a genus zero hyperbolic surface with one cusp equivalent to i∞ and two elliptic fixed points equivalent to z = i and z = eiπ/3 respectively. Functions defined on X1 are usually referred to as modular forms and depending on their behavior at the cusp, such functions are either weakly holomorphic

E-mail address: [email protected]. https://doi.org/10.1016/j.jnt.2018.03.019 0022-314X/© 2018 Elsevier Inc. All rights reserved.

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(if they have a pole at the cusp) or holomorphic. The latter ones are called cusp forms if they vanish at the cusp. The holomorphic Eisenstein series are the modular forms that generate the vector space of fixed even weight k holomorphic functions on X1 , namely functions satisfying f (γz) = (cz + d)k f (z) for all γ(z) = (az + b)/(cz + d) ∈ PSL(2, Z). For even k ≥ 4, the holomorphic Eisenstein series of weight k are given by 

Ek (z) =

−k

(cz + d)

 ,

γ∈Γ∞ \PSL(2,Z)

γ=

∗ ∗ c d

 (1.1)



  1 n : n ∈ Z . The smallest weight for which the vector 0 1 space of holomorphic forms is non-trivial is 4. The Eisenstein series E4 (z) and E6 (z) are the generators for the graded ring of holomorphic modular forms. The smallest weight for which there exists a cusp form is 12. The modular discriminant Δ(z) given by where z ∈ H and Γ∞ =

Δ(z) =

E43 (z)

−

E62 (z)

= η(z)

24

=q

∞ 

(1 − q n )24 ,

q = e2πiz

(1.2)

n=1

is the unique (up to multiplication by a constant) weight 12 cusp form. The holomorphic Eisenstein series can be used to construct the field of rational functions (weakly holomorphic forms) on X1 . In particular, the field of automorphic forms (i.e. weight 0 weakly holomorphic) is generated by the Klein j-invariant which is a rational function in E4 (z) and E6 (z), namely j(z) =

123 E43 (z) . Δ(z)

(1.3)

The Fourier expansion of the j-invariant at the cusp i∞ has the following form j(z) =

1 + 744 + 196884q + 21493760q 2 + O(q 3 ) as q → 0 , q

(1.4) √

with the coefficients satisfying the bound of the form cn = O(n−3/4 e4π n ) as n approaches infinity (see [Ra 38]). There is a rather vast literature on the subject of the j-invariant, but for the sake of brevity of this article, we leave them out, referring the reader to [Ga 06]. In a series of recent articles ([JST 16a], [JST 16b], [JST 16c], [JST 18]), the authors consider analogs to the j-invariant for certain arithmetic subgroups Γ0 (N )+ /{±1}, having square-free positive integer level N . Denote by X0 (N )+ the arithmetic surface obtained via the action of the group on the upper-half plane H. For the genus 0 such surfaces, there is only one generator jN (z) for the function field of automorphic forms. When the surface has genus g with g ≥ 1, the function field on X0 (N )+ has two generators xN (z) and yN (z). At least on surfaces of genus g up to g ≤ 3 the q-expansions of the generators have integer coefficients once the leading coefficients are normalized to 1.

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Furthermore the orders of the poles of the generators at i∞ are at most g + 2. In the process, they construct certain weakly holomorphic modular forms which are then used to express the generators. In [JST 16c] and [JST 18], the authors compare and identify exact algebraic results with numerical approximations of generators at certain algebraic numbers in the setting of genus 0 and 1 groups. In this article, we analytically address the question of convergence of such series. In particular, we establish effective and possibly practical bounds for the coefficients in the q-expansions of the functions used to construct the generators of the function field of X0 (N )+ . The organization of this article is as follows. After recalling some background material in Section 2, we will prove a series of lemmas in Section 3, leading to the main theorem in Section 4. We conclude with a numerical example in Section 5 where we show how the bounds in Theorem 6 may be applied to a particular generator for the function field of rational functions at an imaginary quadratic number. 2. Preliminary material In this section we recall some background material regarding the theory of modular forms on the arithmetic groups Γ0 (N )+ . For any integer N ≥ 1, consider arithmetic subgroups Γ0 (N )+ of level N defined by  +

Γ0 (N ) =

1 √ e



a c

b d



 : a, b, c, d, e ∈ Z, ad − bc = e, e|N, e|a, e|d, N |c ,

noting that Γ0 (1)+ = SL(2, R). The choice of this particular subgroup is due to its moonshine like nature, i.e. Γ0 (N )+ contains the congruence subgroup Γ 0 (N ) =    1 n a b ∈ SL(2, Z) : N |c and any of its horizontal translations of the form ± 0 1 c d + + with n ∈ Z (cf. [Ga 06]). Denote by Γ0 (N ) = Γ0 (N ) /{±1} which may be viewed as  the subgroup of linear rational maps of the form f (z) = (az + b)/(cz + d) with a b ∈ Γ0 (N )+ . The action of Γ0 (N )+ on the upper-half plane H yields the quoc d tient space X0 (N )+ . Throughout this paper we assume a square-free level and write N = p1 · · · pr , with pi ’s primes. In this case X0 (N )+ is a hyperbolic Riemann surface with only one cusp, which may always be taken at i∞. (N ) For any even integer k ≥ 4, denote by Ek (z) the holomorphic Eisenstein series of weight k for the group Γ0 (N )+ , which is given by (N ) Ek (z)

=



(cz + d)

−k

 ,

γ=

∗ ∗ c d

 .

(2.1)

γ∈Γ∞ \Γ0 (N )+

From [JST 16a] we recall that Eisenstein series in (2.1) may be written as linear combinations of Eisenstein series for PSL(2, R) given in (1.1), namely

D. Garbin / Journal of Number Theory 191 (2018) 384–395

(N )

Ek

(z) =

387

 1 v k/2 Ek (vz) , σk/2 (N )

(2.2)

v|N

 with σk (N ) = v|N v k denoting the k-th power divisor function. Using (2.2) together with the q-expansion for the holomorphic Eisenstein series for the full modular group (see for instance [Se 73]), yields the Fourier expansion for Eisenstein series on Γ0 (N )+ (N ) Ek (z)

 1 = v k/2 σk/2 (N )



∞ 2k  1− σk−1 (n)q vn Bk n=1

v|N

,

(2.3)

where Bk denotes the k-th Bernoulli number. Aside from holomorphic Eisenstein series, at each cusp of a hyperbolic Riemann surface (here i∞ only) there is a real-analytic and non-holomorphic Eisenstein series which we denote here by Epar,∞ (z, s) which plays an important role in theory of automorphic forms. For z ∈ H and Re s > 1, the series is given by 

Epar,∞ (z, s) =

(Im ηz)s .

η∈Γ∞ \Γ0 (N )+

The constant coefficient in the Laurent expansion of Epar,∞ (z, s) about s = 1 involves the logarithm of the Dedekind η function which itself may be rewritten as an algebraic function of holomorphic Eisenstein series as in (1.2). In [JST 16a], the Kronecker limit formula for Γ0 (N )+ is studied, resulting into a Kronecker limit function ΔN (z), given by ⎛ ΔN (z) = ⎝



⎞ N η(vz)⎠

,

(2.4)

v|N

with the constant N defined as  N = 21−r lcm

4, 2r−1

24 (24, σ(N ))

 ,

(2.5)

where lcm denotes the least common multiple and σ(N ) = σ1 (N ) – the sum of divisors of N . The function ΔN (z) is a weight kN = 2r−1 N holomorphic form on Γ0 (N )+ vanishing only at the cusp. In fact, it is the smallest weight such form and the analog to + the modular discriminant for PSL(2, Z), noting that Δ(z) = Δ1 (z). If N is such that XN has genus zero or one, the values of kN and N are given in Tables 3 and 4 of [JST 16a]. In [JST 16a] the authors construct automorphic forms (i.e. weight 0 weakly modular functions with pole at i∞) given by Fb (z) =

bν   M (N ) (z) Em ΔN (z) ν ν

where

 ν

bν mν = M kN

and b = (b1 , . . .) .

388

D. Garbin / Journal of Number Theory 191 (2018) 384–395

In the case of genus 0, the holomorphic function field has transcendence degree one, hence there is only one generator jN (z), while in the case of genus g ≥ 1 the holomorphic function field has transcendence degree two which is manifested in two generators xN (z) and yN (z). In all instances up to at least g ≤ 3, the generators are finite linear combinations of functions of the form Fb (z). 3. Bounds on coefficients of holomorphic Eisenstein series In this section we prove a series of lemmas in order to compute bounds on the coefficients in the q-expansions for functions realized as weighted products of holomorphic Eisenstein series for the group Γ0 (N )+ . Lemma 1. Let f1 , f2 , . . . , fH : H → C be a finite set of functions that are invariant under ∞ (h) z → z + 1 and whose corresponding q-expansions n=0 an q n for each h = 1, . . . , H converge absolutely and uniformly on compacta for z ∈ H. Then for the coefficient of q n of the product of these functions defined by ∞ 

n

cn q =

n=0

H 



∞ 

n a(h) n q

,

n=0

h=1

we have the bound |cn | ≤ (n + 1)H−1

H  h=1

(h)

(sup |aj |) .

(3.1)

j≤n

(h)

(h)

Furthermore, if for all h and n, we have that |an | ≤ bn , then |cn | ≤ (n + 1)

H−1

H   h=1

 (h) sup bj j≤n

.

(3.2)

Proof. Let’s consider the case when H = 2 and note that the product converges on the same domain on which the q-expansions of f1 and f2 converge. Next, we can write cn =

n 

(1) (2)

aj an−j ,

j=0

from which the bound is immediate. For H > 2, one can proceed by induction to prove (3.1). The proof of (3.2) follows immediately. 2 Lemma 2. Let s = M N σ(N )/24 and K = M N 2r . Then for any n ≥ −s, the coefficients cn in the q-expansion of (ΔN (z))−M satisfy the bound

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389

|cn | ≤ (n + s + 1)K−1 p(n + s)K , where p(n + s) is the partition function. Proof. Directly from the definition of the Kronecker limit function ΔN (z) in (2.4) and the infinite product for the Dedekind η function in (1.2), we can write ⎛ ΔN (z)M = ⎝



⎞ M N η(vz)⎠

⎛ = q M N σ(N )/24 ⎝

∞ 

⎞M N (1 − q vn )⎠

.

v|N n=1

v|N

Setting s = M N σ(N )/24, it immediately follows that qs =  ΔN (z)M

1

M N .

∞ v|N

vn n=1 (1 − q )

As noted in [JST 16a], page 310, the integer N σ(N ) is divisible by 24, implying that s is integer as well. Recalling that ∞  1 = p(n)q n , n) (1 − q n=1 n=0

∞ allows us to write

 qs = M ΔN (z) v|N



∞ 

M N p(n)q

vn

.

(3.3)

n=0

Since N = p1 · · · pr , there are 2r factors in the product over v | N . We then apply (3.1) with H = M N 2r , to get an upper bound for the coefficient of q n of (3.3). Clearly, p(n) ≤ p(vn) for any v ≥ 1, so then using (3.2), we conclude that the coefficient cn of q n in the right-hand side of (3.3) satisfies the bound cn ≤ (n + 1)K−1 p(n)K with K = M N 2r . The result follows by multiplying (3.3) by q −s , so then the coefficient of qn in (ΔN (z))−M is equal to the coefficient of q n+s in (3.3). 2 Lemma 3. For positive even k, denote by Bk the k-th Bernoulli number. Then we have the following bounds  2k/|Bk | <

1, 504,

if k ≥ 18 . if k ≥ 2

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390

Proof. For convenience, let us write k = 2h. We recall the relation between Bernoulli numbers and the values of the Riemann zeta function at positive even integers, namely B2h =

(−1)h+1 2(2h)! ζQ (2h) (2π)2h

for all integers h ≥ 1. Also, from Stirling’s formula, one has the lower bound √

2π(2h)2h+1/2 e−2h ≤ (2h)!

again valid for all integers h ≥ 1. Using the trivial bound that ζQ (2h) > 1, for all h ≥ 1 we get that √   4h 2k h πe 2h √  πe 2h−1/2 = <√ = e . |Bk | |B2h | h π h

(3.4)

If h ≥ 10, then it is elementary to show that the right-hand side of (3.4) is less than one. A simple computation shows the same is true for h = 9. These h values correspond to k ≥ 18. If 1 ≤ h < 9, one finds that the largest value of the right-hand side of (3.4) is just below 504, attained for h = 3. 2 Lemma 4. For any n ≥ 0, the n-th coefficient in the q-expansion of the weight k holo(N ) morphic Eisenstein series Ek (z) satisfies the bound |cn | < 504σk−1 (n) ≤ 504nk . Proof. With N = p1 · · · pr square-free, starting with formula (2.3), we can write the (N ) holomorphic Eisenstein series Ek (z) as follows (N ) Ek (z)

1 = σk/2 (N ) +

2

... +

∞ 2k  1− σk−1 (n)q n Bk n=1



 r



2k 1− Bk

k/2

(pi1 pi2 )

i1 ,i2

 r  r

∞ 

+

 r 1

σk−1 (n)q

k/2 pi1

i1

pi1 pi2 n

∞ 2k  1− σk−1 (n)q pi1 n Bk n=1

+ ...

n=1

(pi1 · · · pir )

k/2

i1 ,...,ir

∞ 2k  1− σk−1 (n)q pi1 ···pir n Bk n=1

The zeroth coefficient is equal to (1/σk/2 (N )) (N )

Ek



(z) = 1 −

 v|N

 .

v k/2 = 1. So then we can write

∞  2k αn q n σk/2 (N )Bk n=1

(3.5)

with αn = σk−1 (n) when p1 , . . . , pr  n; when exactly j out of r primes divide n, we have

D. Garbin / Journal of Number Theory 191 (2018) 384–395



αn =

391

v k/2 σk−1 (n/v) .

v|pi1 ···pij

If we use all primes from the factorization of N together with the fact that σk−1 (n/v) ≤ σk−1 (n) we can major αn for any n as follows αn ≤



v k/2 σk−1 (n) = σk/2 (N )σk−1 (n) .

v|N

Together with Lemma 3, the coefficient cn with n > 0 in (3.5) is bounded by |cn | =

2kαn < 504σk−1 (n) ≤ 504nk , σk/2 (N )|Bk |

which completes the proof. 2 Lemma 5. Let G(z) =

 (N )  (Emk (z))bk where bk mk = M kN where each mk ≥ 4 is an k

k

even integer. Then for any n ≥ 0, the coefficient of cn in the q-expansion G(z) satisfies the bound |cn | ≤ 504M kN /4 (n + 1)5M kN /4−1 . Proof. Since each mk ≥ 4, there are at most M kN /4 terms in the product, meaning  bk ≤ M kN /4. From Lemma 1 and Lemma 4, we then get that the upper bound for the coefficient of q n is |cn | ≤ (n + 1)M kN /4−1



(504nmk )bk ≤ 504M kN /4 (n + 1)5M kN /4−1 ,

k

as claimed.

2

4. Main Theorem In this section we use all the results from Section 3 in order to derive bounds on the coefficients for weakly modular forms of weight 0 for the group Γ0 (N )+ . Such modular forms are used to construct generators for the field of rational functions on the surfaces X0 (N )+ for various levels N . Theorem 6. Let N = p1 · · · pr square-free, N given by the formula (2.5), s = M N σ(N )/24, and K = M N 2r . Consider the function fN (z) =

 b

Let C∞ = sup |cb |.

cb Fb (z).

392

D. Garbin / Journal of Number Theory 191 (2018) 384–395

i) Assume each Fb in the sum has the same power of ΔN in the denominator, say M .  Let AM denote the number of solutions to ν bν mν = M kN . Then the coefficient cn in the q-expansion of fN is bounded by |cn | ≤ C∞ AM · 504M kN /4 (n + 1)5M kN /4 (n + s + 1)K−1 p(n + s)K . ii) Assume each Fb in the sum has the power of ΔN in the denominator less than or  equal to M . Let BM denote the number of solutions to ν bν mν = M t for any 1 ≤ t ≤ kN . Then the coefficient cn in the q-expansion of fN is bounded by |cn | ≤ C∞ BM · 504M kN /4 (n + 1)5M kN /4 (n + s + 1)K−1 p(n + s)K Proof. Use the previous lemmas. 2 √  Remark 7. Using that p(n) ≤ ec n /n3/4 where c = π 2/3 (see [Prib 09]), we can get the main result in terms of elementary functions. In this manner, the bounds in the above theorem are effective.

Remark 8. The values for the constants AM and BM may be computed using the partition function. We will do so in one particular instance in Section 5.  5. Example: N = 37 and z = i 3/37 In the case N = 37, the surface X0 (37)+ has genus 1 and consequently 2 generators x37 (z) and y37 (z) for its function field of rational functions. In particular, x37 (z) has the following q-expansion x37 (z) = q −2 + 2q −1 +

∞ 

cn q n

(5.1)

n=1

 (see Table 5 in [JST 16a]). The value x37 (i 3/37) is a root of the polynomial p37,3 (x) = −4107 − 2738x − 592x2 − 37x3 + x4 namely,  x37

i

3 37



1 = 4

    √ √ 37 + 37 9 + 158 + 26 37 ,

(5.2)

(see Example 13 of [JST 18]). In establishing (5.2), one computes the partial sum for the series in (5.1) and compares its value against the 4 roots of the polynomial above. In this direction, we can apply Theorem 6 above to the function x37 (z) in order to determine the magnitude of the tail, hence establishing the number of coefficients that are needed in (5.1) so that the partial sum approximation is close enough to the limit of the series. The coefficients cb can be found on http://www.efsa.unsa.ba/~lejla.smajlovic/ for any square-free level subject to the constraint that the genus is less than or equal to 3. We

D. Garbin / Journal of Number Theory 191 (2018) 384–395

393

have downloaded the coefficients for the specific case of x37 and find by inspection that C∞ ≤ e11.706 . The algorithm employed in computing x37 (z) completes successfully in M = 4 iterations (see Section 9 in [JST 16a]). Other values needed are kN = 12 for the weight of Δ37 (z) which then implies that N = 12 and K = 96. To determine BM , the  number of solutions to ν bν mν = 4t for any 1 ≤ t ≤ 12, we reason as follows. Since the weights mν ≥ 4 are even, we reduce the problem to determining the number of solutions  to ν bν (mν /2) ∈ {2, 4, . . . , 22, 24}. For each even number 2 ≤ t ≤ 24, the numbers bν are positive integers and the numbers mν /2 are integers between 2 and t. Consequently,  the number of solutions to ν bν (mν /2) = t corresponds to the number of partitions of t with integers excluding 1, namely p(t) − p(t − 1). This allows us to compute BM as follows 24 

BM =

[p(t) − p(t − 1)] = 891 .

t=2, t even

Using these values  in the context of Theorem 6 part ii) and Remark 7, we can write for the tail of x37 (i 3/37)   ∞     −2π 3/37n  cn e     n=n0

≤

∞ 

e11.702 · 891 · 50412 ·

n=n0

(n + 1)60 (n + 77)95 96π2/3√n+76−2π3/37n ·e , (5.3) (n + 76)72

with n0 > 0. We proceed by analyzing the terms in the right-hand side of (5.3) into three: the constant part v1 , the rational part v2 (n), and the exponential part v3 (n). For the constant part v1 we have v1 = e11.702 · 891 · 50412 < 3 · 1040 .

(5.4)

For the rational part we have  v2 (n) = (n + 1)60 (n + 76)23 1 +

1 n + 76

95 .

If we assume that n ≥ 1000, then n + 1 = (1000n + 1000)/1000 ≤ (1000n + n)/1000 = 1.001n and similarly n + 76 ≤ 1.076n, leading to the following bound for the rational part of (5.3) 

v2 (n) ≤ (1.001n)

60

· (1.076n)

23

1 · 1+ 1076

95 < 7n83 .

(5.5)

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Table 1  Estimates for size of the tail of x37 (i 3/37) starting with n0 . n0

22100

23100

24100

25100

26100

27100

Tail bound

1.53 · 10−242

2.20 · 10−255

5.52 · 10−268

2.35 · 10−280

1.64 · 10−292

1.83 · 10−304

For the exponential part, if we assume that n ≥ 1000, then we can write v3 (n) = e96π



 √ 2/3 n+76−2π 3/37n

≤ e96π



  √ 2/3 1076/1000 n−2π 3/37n

√

= e−κ

n

(5.6)

   √ where κ = −(96π 2/3 1076/1000−2π 3/37 n). For reasons of convergence, we must have that κ > 0 which then requires that ⎡

96π n≥⎢ ⎢ ⎢

2 ⎤   2/3 1076/1000 ⎥  ⎥ = 20384 . 2π 3/37 ⎥

Collecting all the bounds in (5.4), (5.5), and (5.6), we can write for any n0 ≥ 20384   ∞ ∞ ∞    √  √ dt  −2π 3/37n  40 83 −κ n 40 < 3 · 10 cn e 7n e ≤ 21 · 10 e−κ t t84     t n=n0

n=n0

=

<

42 · 1040 κ168

∞

n0

e−t t168

√ κ n0

dt t

√ 1042 Γ(168, κ n0 ) , 168 κ

(5.7) ∞

where we denote for a ≥ 0 the incomplete gamma function Γ(s, a) = a e−x xs−1 dx. We finish this example by looking at some numerical estimates. For all n0 ≥ 22100, it follows κ in (5.7), the tail for the q-expansion that −κ < −10. If we use 10 in place of √ of x37 (i 3/37) is bounded by 10−126 Γ(168, 10 n0 ). The size of the tail for values of n0 ≥ 22100 are recorded in Table 1. Remark 9. The bounds obtained in Theorem 6 are not only effective, but possibly practical in the context of the numerical computations conducted in articles [JST 16c] and [JST 18]. Take for  instance the above example. There we evaluate bounds for x37(N ) at the point z = i 3/37 whose imaginary part is very close to 0. This in turn, requires a minimum rank n0 of at least 22100. For points in the upper-half plane with larger imaginary parts, the corresponding rank n0 would be lower. Remark 10. The bounds dictated by Theorem 6 could be improved on a case by case basis. Each case requires knowledge of the weights of the Eisenstein series involved in the expression of the function fN (z). Furthermore, one needs to revisit Lemma 4 and

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replace the value 504 in the bound involving the Bernoulli number B6 associated to the q-expansion of the holomorphic Eisenstein series of weight 6, by the appropriate and smaller bounds corresponding to each of the other lower weights Eisenstein series involved. In the case that all the included Eisenstein series have weights at least 18, then the value 504M kN /4 in Theorem 6 is immediately replaced by 1, albeit in actual computations a value in the interval (0, 1) would be used: The larger the weight the closer the value would be to 0. Acknowledgments We are grateful to Jay Jorgenson for enlightening discussions pertinent to this article. We also expressed our gratitude to the anonymous referee of this article for all the helpful comments, in particular those regarding the computations carried in Section 5. References [Ga 06] [JST 16a] [JST 16b] [JST 16c] [JST 18]

[Prib 09] [Ra 38] [Se 73]

T. Gannon, Monstrous moonshine: the first twenty-five years, Bull. Lond. Math. Soc. 38 (2006) 1–33. J. Jorgenson, L. Smajlović, H. Then, Kronecker’s limit formula, holomorphic modular functions, and q-expansions on certain arithmetic groups, Exp. Math. 25 (3) (2016) 295–319. J. Jorgenson, L. Smajlović, H. Then, Certain aspects of holomorphic function theory on some genus-zero arithmetic groups, LMS J. Comput. Math. 19 (2) (2016) 360–381. J. Jorgenson, L. Smajlović, H. Then, The Hauptmodul at elliptic points of certain arithmetic groups, arXiv:1602.07426, submitted for publication. J. Jorgenson, L. Smajlović, H. Then, On the evaluation of singular invariants for canonical generators of certain genus one arithmetic groups, Exp. Math. (2018), https://doi.org/10. 1080/10586458.2017.1422161, in press. W. Pribitkin, Simple upper bounds for partition functions, Ramanujan J. 18 (2009) 113–119. H. Rademacher, The Fourier coefficients of the modular invariant J(τ ), Amer. J. Math. 60 (2) (1938) 501–512. J.-P. Serre, A Course in Arithmetic, Graduate Text in Mathematics, vol. 7, Springer-Verlag, New York, 1973, ix+118 pp.