Math. J. Okayama Univ. 59 (2017), 71–79

EVALUATION OF CONVOLUTION SUMS AND SOMEREMARKS ON CUSP FORMS OF WEIGHT 4 AND LEVEL
12B. Ramakrishhan and Brundaban Sahu
Abstract. In this note, we evaluate certain convolution sums and makesome remarks about the Fourier coefficients of cusp forms of weight 4for Γ0 (12). We express the normalized newform of weight 4 on Γ0 (12) asa linear combination of the (quasimodular) Eisenstein series (of weight2) E2 (dz), d|12 and their derivatives. Now, by comparing the workof Alaca-Alaca-Williams [1] with our results, as a consequence, we express the coefficients c1,12 (n) and c3,4 (n) that appear in [1, Eqs.(2.7)
and (2.12)] in terms of linear combination of the Fourier coefficients ofnewforms of weight 4 on Γ0 (6) and Γ0 (12). The properties of c1,12 (n)and c3,4 (n) that are derived in [1] now follow from the properties of theFourier coefficients of the newforms mentioned above. We also express
the newforms as a linear combination of certain eta-quotients and obtainan identity involving eta-quotients.

1. Introduction
Let N, Z, Q be the set of positive integers, integers, rational numbers
respectively. For k ∈ N and n ∈ N, we define σk (n) to be the sum of the kth
powers of the positive divisors of n. We write σ(n) for σ1 (n) and we assume
that σk (x) = 0 for a rational number x which is not an integer. Following[14, 11], for n, N ∈ N, we define WN (n) as follows.
X(1.1)WN (n) =σ(m)σ(n − N m).
m∈N,m<n/N

Also, following [1], we define Wa,b (n) for a, b ∈ N byX(1.2)
Wa,b (n) :=σ(l)σ(m).
l,m∈Nal+bm=nNote that W1,N (n) = WN,1 (n) = WN (n). These type of sums were evaluatedas early as the 19th century. For example, the sum W1 (n) was evaluated by
Besge, Glaisher and Ramanujan [4, 5, 10].
In [11], E. Royer used the structure of the space of quasimodular forms
to evaluate the convolution sums WN (n) for 1 ≤ N ≤ 14, N 6= 12. In
Mathematics Subject Classification. Primary 11F11, 11A25; Secondary 11E20, 11E25,11F20.
Key words and phrases. convolution sums of the divisor function, Fourier coeffificients,
newforms of integral weight.
71

72

B. RAMAKRISHHAN AND B. SAHU[1], A. Alaca, S. Alaca and K. S. Williams have evaluated W12 using a
different method and in [9], we evaluated the convolution sum W15 using
the theory of quasimodular forms. Recently, there have been many works
on the evalution of these sums starting with the work of Huard et. al [6]and we refer to [15, Table 1] for a recent update. We also refer to the bookof Williams [13], which gives a beautiful history about the problem and its
elementary methods.
In this note we evaluate the convolution sums W12 (n) and W3,4 (n) using
the theory of quasimodular forms. Though these convolution sums have been
evaluated earlier by Alaca-Alaca-Williams [1] by using a different method,
the purpose of our evaluation is that we obtain certain interesting propertiesof the Fourier coefficients of certain cusp forms when we compare the tworesults. More precisely, in [1], they expressed W12 (n) and W3,4 (n) in termsof the divisor functions and coefficients of certain q-series given by c1,12 (n)and c3,4 (n) (see [1, Eqs. (2,7) and (2.12)]). They observed some propertiesof c1,12 (n) and c3,4 (n) which implies that the sums W12 (n) and W3,4 (n)have elementary evaluation for n = 2r 3s . Our method in this note gives
the expressions for W12 (n) and W3,4 (n) in terms of divisor functions andFourier coefficients of newforms of weight 4 on Γ0 (6) and Γ0 (12). Thus, bycomparing our results with the result of Alaca-Alaca-Williams as mentioned
above, we get expressions for c1,12 (n) and c3,4 (n) in terms of the Fourier
coefficients τ4,6 (n) and τ4,12 (n) of the newforms of weight 4 with levels 6and 12 respectively. Since the newforms are Hecke eigenforms with respect
the Hecke operators Up for the primes dividing the level (here p = 2, 3), therequired properties (elementary evaluation when n = 2r 3s ) of c1,12 (n) andc3,4 (n) follow from the properties of τ4,6 (n) and τ4,12 (n) (see Remark 3).
For the basic theory of newforms we refer to [3]. We also get several otherconsequences, which are listed below.
(i) In Remark 1, we express the newform ∆4,12 (z) as a linear combinationof the (quasimodular) Eisenstein series (of weight 2) E2 (dz), d|12 and theirderivatives.
(ii) In Remark 2 we express the newform ∆4,12 (z) and the oldform ∆4,6 (z) +4∆4,6 (2z) as a linear combination of certain eta-quotients and also obtainan identity involving some eta-quotients. We express c1,12 (n) and c3,4 (n) interms of the Fourier coefficients of the newforms ∆4,6 (z) and ∆4,12 (z).
(iii) In Remark 4, we give a formula for N4 (n), the number of representationsof a natural number n by the quadratic form x21 + x1 x2 + x22 + x23 + x3 x4 +x24 + 4(x25 + x5 x6 + x26 + x27 + x7 x8 + x28 ) in terms of the divisor functionsand τ4,6 (n), which gives the fact that N4 (n) has elementary evaluation forn = 2r 3s .

CONVOLUTION SUMS73

2. Certain convolution sums
Theorem 2.1. Let n ∈ N. Then,nn
n
1131W12 (n) =σ3 (n) +σ3+σ3+ σ34801602160 3430
9n3n1n+σ(n)σ3+ σ3+−160
6101224 48
  
nn
1n111σ+−− τ4,6 (n) − τ4,6− τ4,12 (n),24 4
1280
20
296


1311nnn
σ3 (n) +σ3+σ3+ σ3W3,4 (n) =4801602160 330  4
9n
3n1nn+σ3+ σ3+−σ160
6101224 163  n
1nn
1
1
1+−
σ− τ4,6 (n) − τ4,6+ τ4,12 (n),24 12480
20296
where τ4,6 (n) and τ4,12 (n) are the nth Fourier coefficients of the cusp forms∆4,6 (z) and ∆4,12 (z), which are the unique normalised newforms of weight
4 on Γ0 (6) and Γ0 (12) respectively. In fact,
(2.1)

222

2

∆4,6 (z) = η (z)η (2z)η (3z)η (6z) :=∞Xτ4,6 (n)q nn=0and(2.2)∆4,12 (z) :=∞Xτ4,12 (n)q n ,n=0

which is a linear combination of certain eta-quotients (see Eq.(3.6)).
Proof. As mentioned in the introduction, we make use of the theory of quasimodular forms to obtain the convolution sums. Since the method is exactly
the same as in [11, 9], we will be brief in giving the proof and leave the details
to the reader. Let Mk (Γ0 (N )) denote the C-vector space of modular forms of≤k/2
weight k on the congruence subgroup Γ0 (N ) and let M̃k(Γ0 (N )) denotethe space of quasimodular forms of weight k, depth ≤ k/2 on Γ0 (N ). The
following structure theorem is the key ingredient in the proof (see [7, 11]).
For an even integer k with k ≥ 2, we havek/2−1(2.3)≤k/2M̃k(Γ0 (N )) =M
j=0

Dj Mk−2j (Γ0 (N )) ⊕ CDk/2−1 E2 ,1 dwhere the differential operator D is defined by D := 2πidz .

74

B. RAMAKRISHHAN AND B. SAHULet Ek (z) denote the normalized Eisenstein series of weight k on SL2 (Z).
For k ≥ 4 it is a modular form of weight k and for k = 2, it is a quasimodular
form of weight 2 and is given byE2 (z) = 1 − 24(2.4)∞
Xσ(n)q n .

n=1When k = 4 and N = 12, Eq. (2.3) gives(2.5)

M̃4≤2 (Γ0 (12)) = M4 (Γ0 (12)) ⊕ DM2 (Γ0 (12)) ⊕ CDE2 .

Using the dimension formula (see for example [8, 12]), it follows that the set{E4 (z), E4 (2z), E4 (3z), E4 (4z), E4 (6z), E4 (12z), ∆4,6 (z), ∆4,6 (2z), ∆4,12 (z)}forms a basis for the space M4 (Γ0 (12)) and the set {Φ1,2 (z), Φ1,3 (z), Φ1,4 (z),Φ1,6 (z), Φ1,12 (z)} forms a basis for the space M2 (Γ0 (12)), where(2.6)Φa,b (z) :=1(bE2 (bz) − aE2 (az)).
b−aTo compute the convolution sum W12 (n), we consider the quasimodular form
E2 (z)E2 (12z) in M̃4≤2 (Γ0 (12)). Therefore, using the structure (2.5) and thebases mentioned above, we have139
2
E4 (z) +E4 (2z) +E4 (3z) + E4 (4z)20020020025271836144+
E4 (6z) + E4 (12z) − ∆4,6 (z) −∆4,6 (2z)200
2555
11−6∆4,12 (z) + DΦ1,12 (z) + DE2 (z).
2In a similar way, to get the convolution sum W3,4 (n), we consider the quasimodular form E2 (3z)E2 (4z) and get the following expression
E2 (z)E2 (12z) =392
1E4 (z) +E4 (2z) +E4 (3z) + E4 (4z)
200
200
200
25
27
1836
144+E4 (6z) + E4 (12z) − ∆4,6 (z) −∆4,6 (2z)
200
25
5
5
3+6∆4,12 (z) + DΦ1,3 (z) + DΦ1,4 (z) + DE2 (z).
2By comparing the n-th Fourier coefficients, we get the required the convolution sums.
E2 (3z)E2 (4z) =3. Remarks on cusp forms of weight 4 and level 12In this section we make several remarks which are consequences of thecomparison of our theorem with the corresponding results of [1].

CONVOLUTION SUMS

75

Remark 1. By taking the difference of the expressions for E2 (z)E2 (12z) andE2 (3z)E2 (4z) given in the proof of the theorem, we get an expression of thenewform ∆4,12 of weight 4 on Γ0 (12) in terms of the Eisenstein series E2 (z)and its derivatives. In fact, subtracting the expressions and simplifying, weget11
1E2 (3z)E2 (4z) − E2 (z)E2 (12z) + DΦ1,12 (z)∆4,12 (z) =
122

(3.1)
3−DΦ1,3 (z) − DΦ1,4 (z) .
2Using the definition of Φa,b (z) from (2.6), we obtain
1
1∆4,12 (z) =E2 (3z)E2 (4z) − E2 (z)E2 (12z) + DE2 (z)
122

(3.2)
3− DE2 (3z) − 2DE2 (4z) + 6DE2 (12z) .
2Remark 2. As mentioned in the introduction, the convolution sums W12 (n)and W3,4 (n) have been evaluated in [1] by a different method, which aregiven below:
n
n
n
1
131W12 (n) =σ3 (n) +σ3+σ3+ σ3480
1602160 34
30
n3n1n9σ3+ σ3+−σ(n)+1606101224 48
  
1nn11−σ−c1,12 (n),+24 412480
n
n
n
1
131W3,4 (n) =σ3 (n) +σ3+σ3+ σ3480160
2160 3
30  4
9n3n
1nn+σ3+ σ3+−σ1606101224 163  1nn1+−
σ−c3,4 (n),24 124
480

where∞
Xn=0

c1,12 (n)q n :=10 η 2 (2z)η 3 (3z)η 3 (4z)η 2 (6z)11η(z)η(12z)+1η 8 (2z)η 8 (6z)
,11 η 2 (z)η 2 (3z)η 2 (4z)η 2 (12z)76

B. RAMAKRISHHAN AND B. SAHU∞
X

c3,4 (n)q n := 10n=0

η 3 (z)η 2 (2z)η 2 (6z)η 3 (12z)η(3z)η(4z)+η 8 (2z)η 8 (6z).
η 2 (z)η 2 (3z)η 2 (4z)η 2 (12z)Comparing the above formulas with the formulas in Theorem 2.1, we can
express c1,12 (n) and c3,4 (n) in terms of τ4,6 (n) and τ4,12 (n):n
624
5(3.3)
c1,12 (n) =τ4,6 (n) + τ4,6+ τ4,12 (n),1111  211n(3.4)
c3,4 (n) = 6τ4,6 (n) + 24τ4,6− 5τ4,12 (n)2Consequently, we can also express τ4,12 (n) in terms of the coefficients c1,12 (n)and c3,4 (n):(3.5)

τ4,12 (n) =111
c1,12 (n) − c3,4 (n).
1010Since c1,12 (n) and c3,4 (n) are nth Fourier coefficients of certain linear combination of eta-quotients (as given above), from the above equation we get an
expression for the newform ∆4,12 (z) as a linear combination of eta-quotients:(3.6) ∆4,12 (z) =η 2 (2z)η 3 (3z)η 3 (4z)η 2 (6z) η 3 (z)η 2 (2z)η 2 (6z)η 3 (12z)−.
η(z)η(12z)η(3z)η(4z)Next, eliminating τ4,12 (n) from the equations (3.3) and (3.4), we get thefollowing.
(3.7)

τ4,6 (n) + 4τ4,6 (n/2) =1(11c1,12 (n) + c3,4 (n)) .
12In other words,(3.8)∆4,6 (z) + 4∆4,6 (2z)=5 η 2 (2z)η 3 (3z)η 3 (4z)η 2 (6z) 1
η 8 (2z)η 8 (6z)+6η(z)η(12z)
6 η 2 (z)η 2 (3z)η 2 (4z)η 2 (12z)5 η 3 (z)η 2 (2z)η 2 (6z)η 3 (12z)+.
6η(3z)η(4z)Note that ∆4,6 (z)+4∆4,6 (2z) is an oldform of weight 4 on Γ0 (12) and we haveexpressed it in terms of a linear combination of eta-quotients. Moreover, if
we use the expression for ∆4,6 (z) in terms of eta-product from equation (2.1),

CONVOLUTION SUMS

77

we get the following identity (after cancelling the factor η 2 (2z)η 2 (6z)):(3.9)
η 2 (z)η 2 (3z) + 4η 2 (4z)η 2 (12z)=5 η 3 (3z)η 3 (4z) 1
η 6 (2z)η 6 (6z)
5 η 3 (z)η 3 (12z)++.
6 η(z)η(12z)
6 η 2 (z)η 2 (3z)η 2 (4z)η 2 (12z) 6 η(3z)η(4z)Remark 3. In [1, §5] (see also [2, §5]), the authors proved some propertiesof the functions c1,12 (n) and c3,4 (n). They showed that12
τ4,6 (n),c3,4 (2n) = 12τ4,6 (n),11−3(3.11)
c1,12 (3n) =c3,4 (n),c3,4 (3n) = −33c1,12 (n).
11Note that in the notation of [1], τ4,6 (n) = c1,6 (n). Since ∆4,6 (z) and ∆4,12 (z)are newforms of weight 4 on Γ0 (6) and Γ0 (12) respectively, these properties
observed in [1] follow easily. The function ∆4,6 (z) is an eigenfunction underthe Hecke operators U2 and U3 with eigenvalues −2 and −3. By the theory of newforms, it is known that the function has eigenvalues ±2 and ±3
with respect to the Hecke operators U2 and U3 respectively and the Fourier
coefficient identity is given by(3.10)

c1,12 (2n) =τ4,6 (np) = ±p τ4,6 (n)(p = 2, 3).

Since τ4,6 (1) = 1, by comparing the Fourier coefficients τ4,6 (2) and τ4,6 (3)we see that the eigenvalues are −2 and −3. Therefore, we get for all n ≥ 1,(3.12)(3.13)

τ4,6 (2n) = −2τ4,6 (n),τ4,6 (3n) = −3τ4,6 (n).

In a similar way, we see that the newform ∆4,12 (z) is an eigenfunction underthe Hecke operators U2 and U3 with eigenvalues 0 and 3. Thus, we have for
all n ≥ 1,(3.14)(3.15)

τ4,12 (2n) = 0,
τ4,12 (3n) = 3τ4,12 (n).

Now the properties (3.10) and (3.11) follow easily using the above propertiesof τ4,6 (n) and τ4,12 (n) together with the identities (3.3) and (3.4). In [1,Eq(5.16)] it is observed that(3.16)

τ4,6 (2r 3s ) = (−1)r+s 2r 3s ,where r, s are non-negative integers. This is also a consequence of the factthat ∆4,6 (z) is a Hecke eigenform with eigenvalues −2 and −3 under theHecke operators U2 and U3 respectively. Since τ4,12 (n) is zero when n is

78

B. RAMAKRISHHAN AND B. SAHU

even, we see that our formulas for W12 (n) and W3,4 (n) have elementaryevaluations when n = 2r 3s .
Remark 4. In [1], the authors used the convolution sums W12 (n) and W3,4 (n)and computed the number N4 (n), which is the number of representations ofa positive integer n by the quadratic form x21 + x1 x2 + x22 + x23 + x3 x4 + x24 +4(x25 + x5 x6 + x26 + x27 + x7 x8 + x28 ) and showed that618  n  54  n  96  n N4 (n) =σ3 (n) + σ3+ σ3+ σ355253549162  n  864  n  99(3.17)
σ3+σ3+ c1,12 (n) + c3,4 (n).
+565121010
Replacing the values c1,12 (n) and c3,4 (n) in terms of τ4,6 (n) and τ4,12 (n) fromequations (3.3) and (3.4), we get
618  n  54  n  96  n N4 (n) =σ3 (n) + σ3+ σ3+ σ35525354n
216
162  n  864  n  54(3.18)
σ3+σ3+ τ4,6 (n) +τ4,6.
+56512552
Notice that the above formula for N4 (n) given in (3.18) depends only on
the values of the divisor functions and the Fourier coefficients τ4,6 (n) of thenewform ∆4,6 (z). Therefore, by using (3.16) it follows that our formulafor N4 (n) has elementary evaluations when n = 2r 3s , where r and s are
non-negative integers.
Acknowledgements
The second author is partially funded by SERB grant SR/FTP/MS053/2012. The authors thank the referee for his/her remarks.
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4l+9m=nB. Ramakrishnan
Harish-Chandra Research Institute
Chhatnag Road, Jhunsi
Allahabad - 211 019 Indiae-mail address: ramki@hri.res.in

Brundaban Sahu
School of Mathematical Sciences
National Institute of Science
Education and Research
PO: Sainik School
Bhubaneswar, Odisha - 751 005 Indiae-mail address: brundaban.sahu@niser.ac.in(Received April 10, 2015 )(Accepted June 17, 2015 )