EXTENSION OF RAMANUJAN’S CONGRUENCES FOR

THE PARTITION FUNCTION MODULO POWERS OF 5

Jeremy Lovejoy and Ken Ono

February 29, 2000.

Abstract. As usual, let p(N) denote the number of partitions of a non-negative integer N. Ramanujan proved (and Watson reproved) that if j ≥ 1 is a positive integer, then for every N we have

p(5^{j}N + β_{5}(j)) ≡ 0 (mod 5^{j})

where β_{5}(j) := 1/24 (mod 5^{j}). Many have questioned the optimality of these congruences. Are there arithmetic subprogressions of 5^{j}N + β_{5}(j), other than those found by Ramanujan, for which the congruence modulo 5^{j} is a congruence modulo 5^{j+1}? Moreover, are there systematic families of such congruences which are themselves analogous to Ramanujan’s original congruences above? We answer both questions in the affirmative by explicitly exhibiting infinitely many such systematic extensions. As an example, we show that if j ≥ 1, then for every non-negative integer N we have

p(5^{j} · 13^{3}N + 5^{j} · 13^{2} + γ(j)) ≡ 0 (mod 5^{j+1})

where

γ(j) := \{ \begin{matrix}(3211 · 5^{j} + 1)/24 if j is odd, \\ (3887 · 5^{j} + 1)/24 if j is even. \end{matrix}

1. Introduction and Statement of Results

A partition of a positive integer n is any non-increasing sequence of positive integers whose sum is n. Let p(n) denote the number of partitions of n. (As usual, we adopt the convention that p(0) = 1 and p(α) = 0 if α ∉ ℕ). Ramanujan’s famous congruences,

1991 Mathematics Subject Classification. Primary 11P83; Secondary 05A17. Key words and phrases. partition function, Ramanujan’s Congruences. The second author is supported by an NSF Early Career grant, an Alfred P. Sloan Foundation Research

Fellowship, and a David and Lucile Packard Research Fellowship.

Typeset by AMS-TEX

which were proved by Atkin, Ramanujan and Watson [2, 3, 14], assert that if j is a positive integer, then

p(5^{j}N + β_{5}(j)) ≡ 0 (mod 5^{j}),  (1)
p(7^{j}N + β_{7}(j)) ≡ 0 (mod 7^{[j/2]+1}),  (2)
p(11^{j}N + β_{11}(j)) ≡ 0 (mod 11^{j})  (3)

for every non-negative integer N where β_{m}(j) := 1/24 (mod m^{j}).

These congruences are quite striking since a cursory examination of values of the partition function fails to reveal further congruences. The mere question as to whether there are infinitely many other congruences of the form

p(AN + B) ≡ 0 (mod M)

had remained open for some time. Although works by the second author [8, 9] have gone some way toward quantifying the rarity of these congruences, it is now known that there are indeed infinitely many such congruences. In fact, Ahlgren and the second author [1, 10] have shown that there are such congruences for every modulus M which is coprime to 6. Unfortunately, these results are not constructive. In fact, to our knowledge no explicit examples of such congruences are known with prime modulus M > 31.

In a similar direction, it is natural to ask whether the moduli in Ramanujan’s congruences (1-3) are optimal. In particular, are there subprogressions, besides those found by Ramanujan, where the known congruence modulo m^{j} is a congruence modulo m^{j+1}? In this paper we revisit Ramanujan’s congruences (1) and answer this question in the affirmative by explicitly exhibiting infinitely many such progressions for each j. Using the ideas found in [10], one can obtain similar extensions of (2) and (3). Unfortunately, these extensions are not palatable.

For convenience, define rational numbers β(j, ℓ) by

β(j, ℓ) := \{ \begin{matrix}\frac{19·5^{j}·ℓ^{2}+1}{24} & if j is odd, \\ \frac{23·5^{j}·ℓ^{2}+1}{24} & if j is even. \end{matrix}  (4)

Notice that β(j, 1) = β_{5}(j). In this notation, we obtain the following systematic families of multiplicative congruences.

Theorem 1. Let ℓ ≥ 7 be prime.

1) If j ≥ 1 is odd, then for every non-negative integer n we have

p(5^{j}ℓ^{2}n + β(j, ℓ)) ≡

≡ (\frac{15}{ℓ}) (1 + ℓ − ℓ^{2}(\frac{−24n − 19}{ℓ})) p(5^{j}n + β_{5}(j)) − ℓp (\frac{5^{j}n}{ℓ^{2}} + β(j, ℓ^{−1})) (mod 5^{j+1}).

2) If j ≥ 2 is even, then for every non-negative integer n we have

p(5^{j}ℓ^{2}n + β(j, ℓ)) ≡

≡ (\frac{15}{ℓ}) (1 + ℓ − (\frac{−24n − 23}{ℓ})) p(5^{j}n + β_{5}(j)) − ℓp (\frac{5^{j}n}{ℓ^{2}} + β(j, ℓ^{−1})) (mod 5^{j+1}).

Using Theorem 1, we obtain two corollaries which reveal extensions of all of Ramanujan’s congruences modulo powers of 5. In both cases, for every positive integer j we construct infinitely many distinct non-trivial subprogressions of the arithmetic progression

5^{j}N + β_{5}(j)

for which Ramanujan’s congruence modulo 5^{j} is a congruence modulo 5^{j+1}.

Corollary 2. Let ℓ ≡ 4 (mod 5) be prime.

1) If j ≥ 1 is odd, let 0 ≤ r, s ≤ ℓ − 1 be integers such that

(i) 24r + 19 ≡ 0 (mod ℓ),
(ii) 24sℓ + 24r + 19 ≢ 0 (mod ℓ^{2}).

2) If j ≥ 2 is even, let 0 ≤ r, s ≤ ℓ − 1 be integers such that

(i) 24r + 23 ≡ 0 (mod ℓ),
(ii) 24sℓ + 24r + 23 ≢ 0 (mod ℓ^{2}).

Then for every non-negative integer N we have

p(5^{j}ℓ^{4}N + 5^{j}ℓ^{3}s + 5^{j}ℓ^{2}r + β(j, ℓ)) ≡ 0 (mod 5^{j+1}).

Corollary 3. Let 7 ≤ ℓ ≡ 3 (mod 5) be prime. If j ≥ 1 is odd (resp. even) and 0 ≤ r ≤ ℓ − 1 is an integer for which (\frac{−24r−19}{ℓ}) = 1 (resp. (\frac{−24r−23}{ℓ}) = −1), then for every non-negative integer N we have

p(5^{j}ℓ^{3}N + 5^{j}ℓ^{2}r + β(j, ℓ)) ≡ 0 (mod 5^{j+1}).

It is easy to see that all of the arithmetic progressions in Corollaries 2 and 3 are not subprogressions of 5^{j+1}N + β_{5}(j + 1). For example, if j is odd in Corollary 3 then

5^{j}ℓ^{3}N + 5^{j}ℓ^{2}r + β(j, ℓ) − β(j + 1, 1) ≡ 0 (mod 5^{j+1})

⟺ 5^{j}ℓ^{3}N + 5^{j}ℓ^{2}r + \frac{19 · 5^{j}ℓ^{2} + 1}{24} − \frac{23 · 5^{j+1} + 1}{24} ≡ 0 (mod 5^{j+1})

⟺ 2N − r − 1 ≡ 0 (mod 5) which is obviously not true for all N.

Examples. Here we illustrate the utility of Corollaries 2 and 3. If j = 1, ℓ = 19, and r = s = 0 in Corollary 2, then we have

p(651605N + 1429) ≡ 0 (mod 25).

Similarly, if j = 1, ℓ = 13 and r = 1 in Corollary 3, then we have

p(10985N + 1514) ≡ 0 (mod 25).

To obtain the infinite family in the abstract simply let ℓ = 13 and r = 1 in Corollary 3. Then for every j ≥ 1 we have

p(5^{j} · 13^{3}N + 5^{j} · 13^{2} + β(j, 13)) ≡ 0 (mod 5^{j+1}).

2. The important observations

In this section we prove two crucial lemmas whose proof implies Theorem 1. As usual, let η(z) denote Dedekind’s eta-function given by the infinite product

η(z) := q^{\frac{1}{24}} ∏_{n=1}^{∞}(1 − q^{n})

where q := e^{2πiz}. We shall study the two modular forms

F (z) := η^{19}(24z) = ∑_{n=1}^{∞} a(n)q^{n} = q^{19} − 19q^{43} + 152q^{67} − …,  (5)

G(z) := η^{23}(24z) := ∑_{n=1}^{∞} b(n)q^{n} = q^{23} − 23q^{47} + 230q^{71} − ….  (6)

It is easy to deduce (see [6]) that F (z) is a cusp form in S_{19/2}(Γ_{0}(576), χ) and that G(z) ∈ S_{23/2}(Γ_{0}(576), χ) where the Nebentypus character χ is the quadratic character defined by

χ(n) := \{ \begin{matrix}1 & if n ≡ ±1 (mod 12), \\ −1 & if n ≡ ±5, (mod 12), \\ 0 & otherwise. \end{matrix}  (7)

The following theorem was first proved by Newman.

Lemma 2.1. (Newman [Th. 1, 9]) If ℓ ≥ 5 is prime and λ_{a}(ℓ) is given by

λ_{a}(ℓ) := a(19ℓ^{2}) + ℓ^{8}(\frac{−57}{ℓ}),

then for every positive integer n we have

λ_{a}(ℓ)a(n) = a(ℓ^{2}n) + ℓ^{8}(\frac{−3n}{ℓ})a(n) + ℓ^{17}a(n/ℓ^{2}).

A similar result is true for the cusp form G(z). In fact, Lemma 2.1 and the next lemma show that F (z) and G(z) are eigenforms of the half integer weight Hecke operators. Recall that if g(z) = ∑_{n=0}^{∞} c(n)q^{n} ∈ M_{λ+ \frac{1}{2}} (Γ_{0}(4N), ψ) is a half integer weight modular form and p ∤ 4N is prime, then the Hecke operator T_{λ}(p^{2}) is given by

g|T_{λ}(p^{2}) := ∑_{n=0}^{∞} (c(p^{2}n) + ψ(p)(\frac{(−1)^{λ}n}{p})p^{λ−1}c(n) + ψ(p^{2})p^{2λ−1}c(n/p^{2})) q^{n}.

Moreover, g is an eigenform if for every prime p ∤ 4N there is a complex number λ_{g}(p) for which

g|T_{λ}(p^{2}) = λ_{g}(p)g.

Lemma 2.2. If ℓ ≥ 5 is prime and λ_{b}(ℓ) is defined by

λ_{b}(ℓ) := b(23ℓ^{2}) + ℓ^{10}(\frac{−69}{ℓ}),

then for every positive integer n we have

λ_{b}(ℓ)b(n) = b(ℓ^{2}n) + ℓ^{10}(\frac{−3n}{ℓ})b(n) + ℓ^{21}b(n/ℓ^{2}).

Proof. This Lemma is equivalent to the assertion that G(z) is an eigenform of the half integer weight Hecke operators T_{11}(p^{2}) on S_{23/2}(Γ_{0}(576), χ) (see [11]). It is routine to check that G(z) is an eigenform of the half-integer weight Hecke operators T_{11}(p^{2}) for every prime p ≥ 5. In particular, using [Prop. 1.3, 4], it suffices to check that G(z) is an eigenform for the Hecke operators T_{11}(p^{2}) for the primes 5 ≤ p < 1104. By the standard dimension counting argument, verifying that G(z) is an eigenform of a Hecke operator T_{11}(p^{2}) reduces to checking that

λ_{b}(p) ∑_{n=1}^{1105} b(n)q^{n} = ∑_{n=1}^{1105} (b(p^{2}n) + p^{10}(\frac{−3n}{ℓ})b(n) + p^{21}b(n/p^{2})) q^{n}.

This computation is easily handled using MAPLE.

□

It turns out that the eigenvalues λ_{a}(ℓ) and λ_{b}(ℓ) satisfy the following convenient congruences. Without such congruences, it seems imposssible to obtain clean extensions of (1).

Theorem 2.3. If ℓ ≥ 5 is prime, then

λ_{a}(ℓ) ≡ λ_{b}(ℓ) ≡ (\frac{15}{ℓ})(1 + ℓ) (mod 5).

To prove this theorem we shall employ some well known facts about modular forms modulo ℓ and the Shimura correspondence [11]. This correspondence is a family of maps which send modular of forms of half-integral weight to those of integer weight. Suppose that f(z) = ∑_{n=1}^{∞} b(n)q^{n} ∈ S_{λ+ \frac{1}{2}} (Γ_{0}(4N), ψ) is an eigenform with λ ≥ 2. If t is any square-free integer, then define A_{t}(n) by

∑_{n=1}^{∞} \frac{A_{t}(n)}{n^{s}} := L(s − λ + 1, ψχ_{−1}^{λ}χ_{t}) · ∑_{n=1}^{∞} \frac{b(tn^{2})}{n^{s}}.

Here χ_{−1} (resp. χ_{t}) is the Kronecker character for ℚ(i) (resp. ℚ(\sqrt{t})). These numbers A_{t}(n) define the Fourier expansion of S_{t,λ}(f(z)), a cusp form

S_{t,λ}(f(z)) := ∑_{n=1}^{∞} A_{t}(n)q^{n}

in S_{2λ}(Γ_{0}(4N), ψ^{2}). For us, the important feature of the Shimura correspondence S_{t,λ} is the fact that it commutes with the Hecke algebra. In other words, if p ∤ 4N is prime, then

S_{t,λ}(f|T_{λ}(p^{2})) = S_{t,λ}(f)|T_{p}^{λ}.  (8)

Here T_{p}^{λ} (resp. T_{λ}(p^{2})) denotes the usual Hecke operator on the space S_{2λ}(Γ_{0}(4N), ψ^{2}) (resp. S_{λ+ \frac{1}{2}} (Γ_{0}(4N), ψ)).

Proof of Theorem 2.3. Lemmas 2.1 and 2.2 are equivalent to the facts that F (z) = η^{19}(24z) ∈ S_{19/2}(Γ_{0}(576), χ) and G(z) = η^{23}(24z) in S_{23/2}(Γ_{0}(576), χ) are eigenforms of the half-integer weight Hecke operators on M_{19/2}(Γ_{0}(576), χ) and M_{23/2}(Γ_{0}(576), χ) respectively.

Now let 𝔉(z) be the eigenform which is the image of F (z) under S_{19,9}, and let 𝔊(z) be the image of G(z) under S_{23,11}.

The first few terms of 𝔉(z) and 𝔊(z) are

𝔉(z) = ∑_{n=1}^{∞} A(n)q^{n} = q − 645150q^{5} − 3974432q^{7} − …,

𝔊(z) = ∑_{n=1}^{∞} B(n)q^{n} = q + 23245050q^{5} + 1322977768q^{7} − …

Since A(1) = B(1) = 1, for every prime ℓ ≥ 5 we have that A(ℓ) (resp. B(ℓ)) is the eigenvalue of 𝔉(z) (resp. 𝔊(z)) with respect to T_{ℓ}^{9} (resp. T_{ℓ}^{11}). Therefore, by the commutativity of the Shimura correspondence (8) we have that A(ℓ) = λ_{a}(ℓ) and B(ℓ) = λ_{b}(ℓ) for every prime ℓ ≥ 5. It turns out that 𝔉(z) is in S_{18}(Γ_{0}(144), χ_{triv}) and that 𝔊(z) is in S_{22}(Γ_{0}(144), χ_{triv}).

If σ_{k}(n) denotes the sum of the kth powers of the positive divisors of an integer n, then it suffices to prove that

𝔉(z) ≡ 𝔊(z) ≡ ∑_{n=1}^{∞} (\frac{60}{n})σ_{1}(n)q^{n} (mod 5).  (9)

This follows from the fact that A(n) = B(n) = 0 if gcd(n, 6) ≠ 1. This fact is easily deduced from the definition of S_{19,9} and S_{23,9} and the fact that a(n) = 0 (resp. b(n) = 0) unless n ≡ 19 (mod 24) (resp. n ≡ 23 (mod 24)).

Recall that the classical weight 6 Eisenstein series E_{6}(z) on SL_{2}(ℤ) is given by

E_{6}(z) = 1 − 504 ∑_{n=1}^{∞} σ_{5}(n)q^{n} ≡ 1 + ∑_{n=1}^{∞} σ_{1}(n)q^{n} (mod 5).

On the basis of Ramanujan’s study of differential operators on modular forms, Swinnerton-Dyer [Lemma 5, 13] proved that if ℓ ≥ 5 is prime and ∑_{n=0}^{∞} c(n)q^{n} is a weight k modular form with integer coefficients, then there is a weight k + l + 1 modular form ∑_{n=0}^{∞} α(n)q^{n} on SL_{2}(ℤ) with integer coefficients whose Fourier expansion satisfies

∑_{n=0}^{∞} α(n)q^{n} ≡ ∑_{n=0}^{∞} nc(n)q^{n} (mod ℓ).

By applying this procedure twice to E_{6}(z) with ℓ = 5, we find that there is a weight 18 modular form H_{0}(z) = ∑_{n=0}^{∞} C(n)q^{n} with respect to SL_{2}(ℤ) such that

H_{0}(z) = ∑_{n=0}^{∞} C(n)q^{n} ≡ ∑_{n=1}^{∞} n^{2}σ_{1}(n)q^{n} (mod 5) ≡ ∑_{n=1}^{∞} (\frac{5}{n})σ_{1}(n)q^{n}.  (10)

If H_{1}(z) is the χ quadratic twist of H_{0}(z), then we have

H_{1}(z) = ∑_{n=0}^{∞} χ(n)C(n)q^{n} ≡ ∑_{n=1}^{∞} (\frac{60}{n})σ_{1}(n)q^{n} (mod 5).

By [III §3 Prop. 17, 6], H_{1}(z) is in the space M_{18}(Γ_{0}(144), χ_{triv}).

Therefore, congruence (9) for 𝔉(z) is equivalent to the assertion that H_{1}(z) ≡ 𝔉(z) (mod 5). By a theorem of Sturm [Th. 1, 12], it suffices to show that

A(n) ≡ (\frac{60}{n})σ_{1}(n) (mod 5)

for every n ≤ 433. A simple computation in MAPLE verifies the congruence (see Appendix for details on computing 𝔉(z)).

Congruence (9) for 𝔊(z) can be handled similarly. Using the fact that the classical Eisenstein series

E_{4}(z) = 1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n} ≡ 1 (mod 5),

it suffices to check that the weight 22 modular form H_{1}(z)E_{4}(z) in M_{22}(Γ_{0}(144), χ_{triv}) satisfies the congruence

H_{1}(z)E_{4}(z) ≡ 𝔊(z) (mod 5).

Using Sturm’s theorem again, this congruence is easily verified in MAPLE by checking that

B(n) ≡ (\frac{60}{n})σ_{1}(n) (mod 5)

for every n ≤ 529 (see Appendix for details on computing 𝔊(z)).

□

RAMANUJAN’S CONGRUENCES MODULO POWERS OF 5 9

3. Proof of Theorem 1 and Corollaries 2 and 3

We begin by recalling the following classical fact regarding the generating functions (see [3, 5, 14]) Theorem 3.1. If j ≥ 1, then the generating function for the numbers p(5^{j}n + β_{5}(j)) is of the form

∑_{n=0}^{∞} p(5^{j}n + β_{5}(j))q^{n} = \{ \begin{matrix}∑_{i≥1} (x_{j,i}q^{i−1} ∏_{n=1}^{∞} \frac{(1−q^{5n})^{6i−1}}{(1−q^{n})^{6i}} ) , & if j is odd, \\ ∑_{i≥1} (x_{j,i}q^{i−1} ∏_{n=1}^{∞} \frac{(1−q^{5n})^{6i}}{(1−q^{n})^{6i+1}} ) , & if j is even, \end{matrix} (11)

where

x_{j,i} = \{ \begin{matrix}5^{j} & if i = 1, \\ 0 (mod 5^{j+1}) & if i ≥ 2. \end{matrix}

The following corollary clarifies the importance of Lemmas 2.1 and 2.2. Corollary 3.2. If j ≥ 1, then for every non-negative integer n we have

p(5^{j}n + β_{5}(j)) ≡ \{ \begin{matrix}5^{j}a(24n + 19) (mod 5^{j+1}) & if j is odd, \\ 5^{j}b(24n + 23) (mod 5^{j+1}) & if j is even. \end{matrix}

Proof. ¿From Theorem 3.1, if j ≥ 1 is odd, then

\frac{1}{5^{j}} ∑_{n=0}^{∞} p(5^{j}n + β_{5}(j))q^{24n+19} ≡ q^{19} ∏_{n=1}^{∞} \frac{(1 − q^{120n})^{5}}{(1 − q^{24n})^{6}} ≡ ∑_{n=0}^{∞} a(n)q^{n} (mod 5).

Similarly, if j ≥ 2 is even, then

\frac{1}{5^{j}} ∑_{n=0}^{∞} p(5^{j}n + β(5, j))q^{24n+23} ≡ q^{23} ∏_{n=1}^{∞} \frac{(1 − q^{120n})^{6}}{(1 − q^{24n})^{7}} ≡ ∑_{n=0}^{∞} b(n)q^{n} (mod 5).

The result follows immediately.

□

Proof of Theorem 1. Lemmas 2.1, 2.2 and Theorem 2.3 imply that

a(ℓ^{2}n) ≡ \{(\frac{15}{ℓ}) (1 + ℓ − ℓ^{2}(\frac{−n}{ℓ}))\} a(n) − ℓa(n/ℓ^{2}) (mod 5)  (12)

and

b(ℓ^{2}n) ≡ \{(\frac{15}{ℓ}) (1 + ℓ − (\frac{−n}{ℓ}))\} b(n) − ℓb(n/ℓ^{2}) (mod 5).  (13)

Congruence (12) follows from the simple observation that (\frac{5}{ℓ}) ≡ ℓ^{2} (mod 5).

Replacing n by 24n + 19 and 24n + 23 in (12) and (13), repectively, and applying Corollary 3.2 immediately establishes the result.

□

We conclude with the proofs of Corollaries 2 and 3.

Proof of Corollary 2. Replace n by Nℓ^{2} + sℓ + r in Theorem 1 and note that

\frac{5^{j}(ℓ^{2}N + sℓ + r)}{ℓ^{2}} + β(j, ℓ^{−1})

is not an integer. Therefore, since (\frac{a}{ℓ}) = 0 if ℓ | a and p(5^{j}n + β_{5}(j)) ≡ 0 (mod 5^{j}), we have that

p(5^{j}ℓ^{4}N + 5^{j}ℓ^{3}s + 5^{j}ℓ^{2}r + β(j, ℓ)) ≡ 0 (mod 5^{j+1}).
□

Proof of Corollary 3. Replace n by ℓN + r and it is easy to see that

\frac{5^{j}(ℓN + r)}{ℓ^{2}} + β(j, ℓ^{−1})

cannot be an integer. Therefore, since p(5^{j}n + β_{5}(j)) ≡ 0 (mod 5^{j}) we have that

p(5^{j}ℓ^{3}N + 5^{j}ℓ^{2}r + β(j, ℓ)) ≡ 0 (mod 5^{j+1}).
□

Appendix

Here we obtain a “closed formula” for the coefficients of the newform 𝔉(z). Computing 𝔊(z) is handled similarly. Let 𝔉_{0}(z) = ∑_{n=1}^{∞} A_{0}(n)q^{n} (resp. 𝔊_{0}(z) = ∑_{n=1}^{∞} B_{0}(n)q^{n}) be the unique newform in the space S_{18}(Γ_{0}(6), χ_{triv}) (resp. S_{22}(Γ_{0}(6), χ_{triv}) whose Fourier expansion are

𝔉_{0}(z) = q − 256q^{2} − 6561q^{3} + 65536q^{4} + 645150q^{5} + 1679616q^{6} + …
𝔊_{0}(z) = q + 1024q^{2} + 59049q^{3} + 1048576q^{4} − 23245050q^{5} + 60466176q^{6} − ….

The newform 𝔉(z) (resp. 𝔊(z)) is the χ quadratic twist of 𝔉_{0}(z) (resp. 𝔊_{0}(z)). In particular, we have that

𝔉(z) = ∑_{n=1}^{∞} A(n)q^{n} = ∑_{gcd(n,6)=1} (\frac{12}{n})A_{0}(n)q^{n},  (14)

𝔊(z) = ∑_{n=1}^{∞} B(n)q^{n} = ∑_{gcd(n,6)=1} (\frac{12}{n})B_{0}(n)q^{n}.  (15)

The proof of Theorem 2.3 requires the first 433 (resp. 529) terms of 𝔉(z) (resp. 𝔊(z)). Define rational numbers D(n) by

∑_{n=1}^{∞} D(n)q^{n} = \frac{−2132029}{4734}η^{29}(z)η^{5}(2z)η(3z)η(6z) − \frac{45171755}{4734}η^{25}(z)η(2z)η^{5}(3z)η^{5}(6z)(16)

− \frac{49149076}{7101}η^{24}(z)η^{6}(2z)η^{6}(6z) − \frac{14062152}{263}η^{20}(z)η^{2}(2z)η^{4}(3z)η^{10}(6z)

+ \frac{204636}{263}η^{18}(z)η^{18}(3z) + (η^{29}(z)η^{5}(2z)η(3z)η(6z) | U(2))

+ (\frac{4539931}{113616}η^{24}(z)η^{6}(2z)η^{6}(6z) − \frac{25504}{2367}η^{25}(z)η(2z)η^{5}(3z)η^{5}(6z)) | U(2).

As usual, the U-operator is defined by

(∑_{n=0}^{∞} b(n)q^{n}) | U(M) := ∑_{n=0}^{∞} b(Mn)q^{n}.

It turns out that if n is coprime to 6, then

D(n) = A_{0}(n) = χ(n)A(n).  (17)

One may use (14) and (17) to compute the first 433 coefficients of the newform 𝔉(z).

Similarly, it is staightforward to obtain 𝔊(z) in terms of eta-products and their images under certain Hecke operators. For brevity we omit the details.

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[13] H. P. F. Swinnerton-Dyer, On ℓ-adic representations and congruences for coefficients of modular forms, Modular functions of one variable III, Springer Lect. Notes in Math. 350 (1973), 1-55.

[14] G. N. Watson, Ramanujan’s vermutung über zerfällungsanzahlen, J. reine angew. math. 179 (1938), 97-128.

Dept. of Math., Penn State University, University Park, Pennsylvania 16802, USA.

   E-mail address: lovejoy@math.psu.edu

Dept. of Math., Penn State University, University Park, Pennsylvania 16802, USA.

   E-mail address: ono@math.psu.edu

Dept. of Math., University of Wisconsin, Madison, Wisconsin 53706

   E-mail address: ono@math.wisc.edu