EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 8, No. 1, 2015, 81-110
ISSN 1307-5543 – www.ejpam.com

Evaluation of Some Convolution Sums by Quasimodular Forms

Barı̧s Kendirli

Department of Mathematics, Art & Sciences Faculty, Fatih University, Istanbul, Turkey

Abstract. We evaluate the convolution sums
∑

l+27m=nσ (l)σ (m) ,
∑

3l+9m=nσ (l)σ (m) ,
∑

l+40m=nσ (l)σ (m) ,
∑

5l+8m=nσ (l)σ (m) ,
∑

4l+10m=nσ (l)σ (m) ,
∑

l+55m=nσ (l)σ (m) ,
∑

5l+11m=nσ (l)σ (m) ,
∑

l+5m=nσ (l)τ2,11 (m) ,
∑

5l+m=nσ (l)τ2,11 (m) ,
∑

11l+5m=nσ(l)τ2,11 (m) ,
∑

55l+m=nσ(l)τ2,11 (m) ,
∑

l+5m=n τ2,11(l)τ2,11 (m) ,

and for all positive integers n using the theory of quasimodular forms, we determine the number of
representations of a positive integer n by the forms

x2
1 + x1 x2 + x2

2 + x2
3 + x3 x4 + x2

4 + 9
�

x2
5 + x5 x6 + x2

6 + x2
7 + x7 x8 + x2

8

�

,

x2
1 + 2x2

2 + x2
3 + 2x2

4 + 5
�

x2
5 + 2x2

6 + x2
7 + 2x2

8

�

,

x2
1 + x1 x2 + 3x2

2 + x2
3 + x3 x4 + 3x2

4 + 5
�

x2
5 + x5 x6 + 3x2

6 + x2
7 + x7 x8 + 3x2

8

�

.

Key Words and Phrases: Quasimodular forms, divisor functions, convolution sums, representation
number

1. Introduction

Let σ(m) be the sum of positive divisors of a positive integer m. It is well known that
divisor function σ appears in a number of remarkable identities, including relationship on
the Riemann zeta function and the Eisenstein series of modular forms. It was studied by
Ramanujan [22], who has found a number of important congruences and identities. It was
also used in the counting of the number of nonisomorphic branched coverings of surfaces of
genus g with a given ramification type σ, and in the orbitwise counting of H(2), see [19]. On
the other hand, the work on representation number r(Q, n) of quadratic forms has been started

Email address: bkendirli@fatih.edu.tr (B. Kendirli)

http://www.ejpam.com 81 c© 2015 EJPAM All rights reserved.



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 82

by Fermat in 1640 on Q = x2 + y2. Later the formula r (Q, n) = 4
�

∑

d|n d is odd (−1)
d−1

2

�

has

been proved by Euler. Afterwards it was advanced by Jacobi, see [11] with the proof of

r (Q, n) = 8

 

∑

d|n4d

d

!

,Q = x2 + y2 + z2 + t2.

It would be nice to obtain such simple formulas for other positive definite quadratic forms so
that we would be able to understand the number of solutions of the equation Q = n for any
positive integer.

Here, we want to study some convolutions of divisor functions

WN (n) :=
∑

m<n/N
m,n∈N

σ(m)σ(n− Nm),

Wa,b(n) :=
∑

al+bm=n
l,m∈N

σ(l)σ(m)

where N , a, b positive integers. Some of them were calculated as early as 19th century. For
example, W1(n) was evaluated by [7], [10], and [22]. The convolution sums WN (n) (for
1 ≤ N ≤ 24 with a few exceptions) and Wa,b(n) for (a, b) ∈ {(2,3) , (3,4) , (3, 8) , (2, 9)} have
been evaluated by using either elementary methods or analytic methods (see [1–3, 5–10, 12,
16–18, 20, 22, 24, 25]).

A different important algebraic method by means of quasimodular forms was introduced
by Royer in [23] and it was applied in [21], in order to get W15,W3,5. The quasimodular forms
was defined in [13]. The number of representations of integers by certain quadratic forms
(see [3, 4, 6, 12, 15, 21, 24, 25]) have been found by evaluation of these convolution sums.
In this article, following the method of Royer [23], and using Magma for the calculations, we
evaluate the following convolution sums

W27 (n) , W3,9 (n) , W40 (n) , W5,8 (n) ,

W20 (n) , W4,5 (n) , W10 (n) , W2,5 (n) ,

W55(n) and W5,11(n),

by using the theory of quasimodular forms. The integers 27, 40,20, 55 are selected, as repre-
sentatives of large classes, p3, q · p3, qp2, qp, respectively. Here p, q different primes.

Although, in general it may be a challenging problem to describe the integer solutions
to a polynomial equation of several variables, by using the above convolution sums we will
determine the formulas for the number of representations of integers of the following positive
definite quadratic forms

Q1 =x2
1 + x1 x2 + x2

2 + x2
3 + x3 x4 + x2

4

+ 9
�

x2
5 + x5 x6 + x2

6 + x2
7 + x7 x8 + x2

8

�

,



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 83

Q2 =x2
1 + 2x2

2 + x2
3 + 2x2

4 + 5
�

x2
5 + 2x2

6 + x2
7 + 2x2

8

�

,

Q3 =x2
1 + x1 x2 + 3x2

2 + x2
3 + x3 x4 + 3x2

4

+ 5
�

x2
5 + x5 x6 + 3x2

6 + x2
7 + x7 x8 + 3x2

8

�

.

These quadratic forms are special cases of the following direct sum of binary quadratic forms
Q

Q+Q+ p2(Q+Q), disc(Q) =− p,

Q+Q+ q(Q+Q), disc(Q) =− p3,

Q+Q+ q(Q+Q), disc(Q) =− p.

So our examples are representatives of large classes of octonary quadratic forms. One can
also obtain the representation number formulas by generalized theta series without using the
convolutions of divisor sums, see [14].

2. Evaluation of Wa,b(n)

In this section, we evaluate the convolution sums

W27 (n), W3,9 (n) , W40 (n) , W5,8 (n) , W20 (n) ,

W4,5 (n), W10 (n) , W2,5 (n) , W55(n) and W5,11(n),

by using the lemma 1.17 in [23]

eM≤2
4 [Γ0(N)] = M4[Γ0(N)]⊕ DM2[Γ0(N)]⊕CDE2,

about the structure of quasimodular forms, where the differential operator D is defined by
D := 1

2πi
d
dz . As an application, we use these convolution sums together with the convolution

sums

W9(n) =
1

216
σ3(n) +

1
27
σ3

�

n
3

�

+
3
8
σ3

�

n
9

�

−
1
36

nσ1(n)

−
1
4

nσ1

�

n
9

�

+
1

24
σ1(n) +

1
24
σ1

�

n
9

�

−
1
54
τ4,9(n).

W3(n) =
1
24
σ3(n) +

3
8
σ3

�

n
3

�

−
1

12
nσ(n)−

1
4

nσ
�

n
3

�

+
1

24
σ(n) +

1
24
σ

�

n
3

�

,

W5(n) =
5

312
σ3(n) +

125
132

σ3

�

n
5

�

−
1
20

nσ(n)−
1
4

nσ
�

n
5

�

+
1

24
σ (n) +

1
24
σ

�

n
5

�

−
1

130
τ4,5(n),

given in [23] to obtain a formula for the number of representations of a positive integer n by
the quadratic forms.



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 84

2.1. Evaluation of W27(n) W3,9(n)

The vector space M4[Γ0(27)] has dimension 12. The linearly independent Eisenstein forms
are E4, E4 (3z) , E4 (9z) , E4 (27z),

Eψ,ψ
4 =

∑

0<d|n

σ
ψ,ψ
3 (n)qn =

∞
∑

n=1

ψ(n)σ3 (n)q
n, Eψ,ψ

4 (3z)

where ψ is the nontrivial Dirichlet character mod 3. There are 4 newforms in S4

�

Γ0 (27)
�

,

∆4,27,1 =q+ 3q2 + · · ·+ 0q27 +O(q28)

∆4,27,2 =q− 3q2 + · · ·+ 0q27 +O(q28)

∆4,27,3 =q+ sq2 + · · ·+ 0q27 +O(q28)

and ∆4,27,4. Here ∆4,27,4 is the Galois conjugate of ∆4,27,3 and s is the zero of x2 − 18, i.e.,
s = 3

p
2. There are 2 old forms,

∆4,9 = (∆(3z))1/3 = q− 8q4 + · · ·+ 0q27 +O(q28),

∆4,9 (3z) in S4[Γ0(27)].
M2

�

Γ0 (27)
�

6 dimensional. There is only one newform

∆2,27 = q− 2q4 + · · ·+ 0q27 +O(q28)

and there are 5 linearly independent Eisenstein Series.

Φ1,3, Eψ,ψ
2 ,Φ1,9,Φ1,27, Eψ,ψ

2 (3z) ,

where

Φa,b =
1

b− a

�

bE2(bz)− aE2(az)
�

Eψ,ψ
2 (z) =

∞
∑

n=1

ψ(n)σ (n)qn.

Now

E2 (27z) E2 (z) =1+
+∞
∑

n=1

�

−24
�

σ(n) +σ
�

n
27

��

+ 576W27(n)
�

qn

W27 (n) =−
1

108
nσ (n) +

1
24
σ (n)−

1
4

nσ
�

n
27

�

+
1
24
σ

�

n
27

�

+
1

1944
σ3 (n) +

1
243

σ3

�

n
3

�

+
1
27
σ3

�

n
9

�

+
3
8
σ3

�

n
27

�

−
1
81
τ4,27,1(n) +

1
972
(−w− 6)τ4,27,3(n) +

1
972
(w− 6)τ4,27,4(n)



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 85

−
2

243
τ4,9(n)−

2
27
τ4,9(

n
3
).

Similarly,

E2 (3z) E2 (9z) =1+
+∞
∑

n=1

�

−24
�

σ(
n
9
) +σ

�

n
3

��

+ 576W3,9(n)
�

qn

=
1
10

E4,3 (z) +
9

10
E4,9 (z) +

4
9

DΦ1,3 +
16
9

DΦ1,9 −
4
9

DE2.

So,

W3,9(n) =−
1
36

nσ
�

n
3

�

+
1
24
σ

�

n
3

�

−
1
12

nσ
�

n
9

�

+
1
24
σ

�

n
9

�

+
1
24
σ3

�

n
3

�

+
3
8
σ3

�

n
9

�

.

2.2. Evaluation of W40(n), W5,8(n), W2,20(n), W4,10(n)

The vector space M4[Γ0(40)] has dimension 22 and is spanned by the 8 linearly indepen-
dent Eisenstein forms

E4, E4,2, E4,4, E4,5, E4,8, E4,10, E4,20, E4,40,

11 old cusp forms

∆4,8 = (∆(2z)∆(4z))1/6 = q− 4q3 + · · ·+ 0q60 +O(q61),

∆4,8 (5z) ,

∆4,10 (z) = q+ 2q2 + · · ·+ (−160)q60 +O(q61),

∆4,10 (2z) ,∆4,10 (4z) ,

∆4,20 (z) = q+ 4q3 + · · ·+ 0q60 +O(q61),

∆4,20 (2z) ,

∆4,5 (z) = (∆(z)∆(5z))1/6 = q− 4q2 + · · ·+ (−80)q60 +O(q61),

∆4,5 (2z) ,∆4,5 (4z) ,∆4,5 (8z) ,

and three newforms

∆4,40,1 (z) =q+ 4q3 + · · ·+ 0q60 +O(q61),

∆4,40,2 (z) =q− 6q3 + · · ·+ 0q60 +O(q61),

∆4,40,3 (z) =q+ 10q3 + · · ·+ 0q60 +O(q61).

The vector space M2[Γ0(40)] has dimension 10. It is spanned by 7 linearly independent
Eisenstein series

Φ1,5,Φ1,10,Φ1,20,Φ1,40,Φ1,2,Φ1,4,Φ1,8,



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 86

2 old forms
∆2,20 = q− 2q3 + · · ·+ 0q60 +O(q61),

∆2,20(2z) and one newform

∆2,40 = q+ q5 + · · ·+ 0q60 +O(q61).

So,

E2 (40z) E2 (z) =1+
+∞
∑

n=1

�

−24
�

σ(
n

40
) +σ (n)

�

+ 576W40(n)
�

qn

=
1

2080
E4 (z) +

3
2080

E4,2 (z) +
3

520
E4,4 (z) +

5
416

E4,5 (z) +
2

65
E4,8 (z)

+
15

416
E4,10 (z) +

15
104

E4,20 +
10
13

E4,40 −
9
2
∆4,40,1(z) −

18
7
∆4,40,3(z)

−
27
14
∆4,8 (z)−

675
14
∆4,8 (5z)−

6
5
∆4,10 (z)−

24
5
∆4,10 (2z)

−
96
5
∆4,10 (4z)−

9
2
∆4,20 (z)− 18∆4,20 (2z)−

378
65
∆4,5 (z)

−
648
13
∆4,5 (2z)−

2592
13
∆4,5 (4z)−

24192
65

∆4,5 (8z) +
117
20

DΦ1,40 (z)

−
3

10
DE2 (z) .

Therefore, we get

W40 (n) =−
1

160
nσ (n) +

1
24
σ (n)−

1
4

nσ
�

n
40

�

+
1

24
σ

�

n
40

�

+
1

4992
σ3 (n) +

1
1664

σ3

�

n
2

�

+
1

416
σ3

�

n
4

�

+
25

4992
σ3

�

n
5

�

+
1
78
σ3

�

n
8

�

+
25

1664
σ3

�

n
10

�

+
25
416

σ3

�

n
20

�

+
25
78
σ3

�

n
40

�

−
1

128
τ4,40,1 (n)−

1
224

τ4,40,3 (n)

−
3

896
τ4,8 (n)−

75
896

τ4,8

�

n
5

�

−
1

480
τ4,10 (n)−

1
120

τ4,10

�

n
2

�

−
1
30
τ4,10

�

n
4

�

−
21

2080
τ4,5 (n)−

9
10

4τ4,5

�

n
2

�

−
9
26
τ4,5

�

n
4

�

−
42
65
τ4,5

�

n
8

�

−
1

128
τ4,20 (n)−

1
32
τ4,20

�

n
2

�

.

Similarly,

E2 (5z) E2 (8z) =1+
+∞
∑

n=1

�

−24
�

σ(
n
8
) +σ

�

n
5

��

+ 576W5,8(n)
�

qn



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 87

=
1

2080
E4 (z) +

3
2080

E4,2(z) +
3

520
E4,4 (z) +

5
416

E4,5 (z) +
2
65

E4,8 (z)

+
15
416

E4,10 (z) +
15

104
E4,20 +

10
13

E4,40 +
9
2
∆4,40,1 (z)−

18
7
∆4,40,3(z)

−
27
14
∆4,8 (z)−

675
14
∆4,8 (5z) +

6
5
∆4,10 (z) +

24
5
∆4,10 (2z)

+
96
5
∆4,10 (4z) +

9
2
∆4,20 (z) + 18∆4,20 (2z)−

378
65
∆4,5 (z)

−
648
13
∆4,5 (2z)−

2592
13
∆4,5 (4z) +

24192
65

∆4,5 (8z) +
3
5

DΦ1,5 (z)

+
21
20

DΦ1,8 (z) +
3
10

DE2 (z) .

So we obtain

W5,8 (n) =−
1

32
nσ
�

n
5

�

+
1
24
σ

�

n
5

�

−
1
20

nσ
�

n
8

�

+
1
24
σ

�

n
8

�

+
1

4992
σ3 (n) +

1
1664

σ3

�

n
2

�

+
1

416
σ3

�

n
4

�

+
25

4992
σ3

�

n
5

�

+
1

78
σ3

�

n
8

�

+
25

1664
σ3

�

n
10

�

+
25
416

σ3

�

n
20

�

+
25
78
σ3

�

n
40

�

+
1

128
τ4,40,1 (n)−

1
224

τ4,40,3 (n)

−
3

896
τ4,8 (n)−

75
896

τ4,8

�

n
5

�

+
1

480
τ4,10 (n) +

1
120

τ4,10

�

n
2

�

+
1
30
τ4,10

�

n
4

�

−
21

2080
τ4,5 (n)−

9
104

τ4,5

�

n
2

�

−
9

26
τ4,5

�

n
4

�

−
42
65
τ4,5

�

n
8

�

+
1

128
τ4,20 (n) +

1
32
τ4,20

�

n
2

�

.

Similarly,

E2 (20z) E2 (2z) =1+
+∞
∑

n=1

�

−24
�

σ(
n

20
) +σ

�

n
2

��

+ 576W20(n)
�

qn

=
1

130
E4,2 (z) +

2
65

E4,4 (z) +
5
26

E4,10 (z)
10
13

E4,20 −
24
5
∆4,10 (2z)

−
432
65
∆4,5 (2z)−

1728
65
∆4,5(4z) +

3
20

DΦ1,2 +
57
20

DΦ1,20 +
3
10

DE2.

Hence we obtain

W2,20 (n) =−
1

80
nσ
�

n
2

�

+
1

24
σ

�

n
2

�

−
1
8

nσ
�

n
20

�

+
1

24
σ

�

n
20

�

+
1

312
σ3

�

n
2

�

+
1
78
σ3

�

n
4

�



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 88

+
25

312
σ3

�

n
10

�

+
25
78
σ3

�

n
20

�

−
1

120
τ4,10

�

n
2

�

−
3

260
τ4,5

�

n
2

�

−
3

65
τ4,5

�

n
4

�

.

Now

E2 (10z) E2 (4z) =1+
+∞
∑

n=1

�

−24
�

σ(
n

10
) +σ

�

n
4

��

+ 576W10,4(n)
�

qn

=
1

130
E4,2 (z) +

2
65

E4,4 (z) +
5
26

E4,10 (z) +
10
13

E4,20 (z) +
24
5
∆4,10 (2z)

−
432
65
∆4,5 (2z)−

1728
65
∆4,5(4z) +

9
20

DΦ1,4 +
27
20

DΦ1,10 +
3
10

DE2.

So

W4,10 (n) =−
1

40
nσ
�

n
4

�

+
1

24
σ

�

n
4

�

−
1

16
nσ
�

n
10

�

+
1

24
σ

�

n
10

�

+
1

312
σ3

�

n
2

�

+
1
78
σ3

�

n
4

�

+
25
312

σ3

�

n
10

�

+
25
78
σ3

�

n
20

�

+
1

120
τ4,10

�

n
2

�

−
3

260
τ4,5

�

n
2

�

−
3
65
τ4,5

�

n
4

�

.

2.3. Evaluation of W20(n), W4,5(n), W2,10(n)

The vector space M4[Γ0(20)] has dimension 12 and is spanned by 6 linearly independent
Eisenstein forms

E4, E4,2, E4,4, E4,5, E4,10, E4,20,

5 old cusp forms
∆4,10 (z) ,∆4,10 (2z) ,∆4,5 (z) ,∆4,5 (2z) ,∆4,5 (4z) ,

and one newform
∆4,20 (z) = q+ 4q3 + · · ·+ 0q60 +O(q61).

The vector space M2[Γ0(20)] has dimension 6 and is spanned by 5 linearly independent
Eisenstein forms Φ1,2,Φ1,4,Φ1,5,Φ1,10,Φ1,20, and one newform ∆2,20. So,

E2 (20z) E2 (z) =1+
+∞
∑

n=1

�

−24
�

σ(
n

20
) +σ (n)

�

+ 576W20(n)
�

qn

=
1

520
E4 (z) +

3
520

E4,2 (z) +
2
65

E4,4 (z) +
5

104
E4,5 (z) +

15
104

E4,10 (z)

+
10
13

E4,20 (z)−
12
5
∆4,10 (z)−

48
5
∆4,10 (2z)− 6∆4,20 (z)

−
576
65
∆4,5 (z)−

720
13
∆4,5 (2z)−

9216
65
∆4,5 (4z) +

57
10

DΦ1,20 (z) +
3
5

DE2 (z) .



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 89

Hence,

W20(n) =−
1
80

nσ (n) +
1
24
σ (n)−

1
4

nσ
�

n
20

�

+
1
24
σ

�

n
20

�

+
1

1248
σ3 (n) +

1
416

σ3

�

n
2

�

+
1

78
σ3

�

n
4

�

+
25

1248
σ3

�

n
5

�

+
25
416

σ3

�

n
10

�

+
25
78
σ3

�

n
20

�

−
1

240
τ4,10 (n)−

1
60
τ4,10

�

n
2

�

−
1

65
τ4,5 (n)−

5
52
τ4,5

�

n
2

�

−
16
65
τ4,5

�

n
4

�

−
1
96
τ4,20 (n) .

Now,

E2 (10z) E2 (2z) =1+
+∞
∑

n=1

�

−24
�

σ(
n

10
) +σ

�

n
2

��

+ 576W2,10(n)
�

qn

=
1

26
E4,2 +

25
26

E4,10 −
288
65
∆4,5 (2z) +

3
10

DΦ1,2 (z) +
27
10

DΦ1,10 (z)

+
3
5

DE2 (z) ,

so

W2,10 =−
1

40
nσ
�

n
2

�

+
1
24
σ

�

n
2

�

−
1
8

nσ
�

n
10

�

+
1

24
σ

�

n
10

�

+
5

312
σ3

�

n
2

�

+
125
312

σ3

�

n
10

�

−
1

130
τ4,5

�

n
2

�

.

Similarly,

E2 (4z) E2 (5z) =1+
+∞
∑

n=1

�

−24
�

σ(
n
5
) +σ

�

n
4

��

+ 576W4,5(n)
�

qn

=
1

520
E4 (z) +

3
520

E4,2 (z) +
2

65
E4,4 (z) +

5
104

E4,5 (z) +
15

104
E4,10 (z)

+
10
13

E4,20 (z) +
12
5
∆4,10 (z) +

48
5
∆4,10 (2z) + 6∆4,20 (z)

−
576
65
∆4,5 (z)−

720
13
∆4,5 (2z)−

9216
65
∆4,5 (4z) +

6
5

DΦ1,5 (z)

+
9
10

DΦ1,4 (z) +
3
5

DE2 (z) .



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 90

Therefore,

W4,5(n) =−
1

20
nσ
�

n
4

�

+
1
24
σ

�

n
4

�

−
1
16

nσ
�

n
5

�

+
1

24
σ

�

n
5

�

+
1

1248
σ3 (n) +

1
416

σ3

�

n
2

�

+
1

78
σ3

�

n
4

�

+
25

1248
σ3

�

n
5

�

+
25

416
σ3

�

n
10

�

+
25
78
σ3

�

n
20

�

+
1

240
τ4,10 (n) +

1
60
τ4,10

�

n
2

�

−
1
65
τ4,5 (n)−

5
52
τ4,5

�

n
2

�

−
16
65
τ4,5

�

n
4

�

+
1
96
τ4,20 (n) .

2.4. Evaluation of W10(n) W2,5(n)

The vector space M4[Γ0(10)] has dimension 7 and is spanned by the linearly independent
Eisenstein forms E4, E4,2, E4,5, E4,10, 2 old cusp forms ∆4,5 (z), ∆4,5 (2z), 1 newform ∆4,10 (z).
The vector space M2[Γ0(10)] has dimension 3, and spanned by Φ1,2,Φ1,5,Φ1,10. Now,

E2 (10z) E2 (z) =1+
+∞
∑

n=1

�

−24
�

σ(
n

10
) +σ (n)

�

+ 576W10(n)
�

qn

=
1

130
E4 +

2
65

E4,2 +
5
26

E4,5 +
10
13

E4,10 −
24
5
∆4,10 (z)−

432
65
∆4,5 (z)

−
1728

65
∆4,5 (2z) +

27
5

DΦ1,10 +
6
5

DE2.

So,

W10(n) =−
1
40

nσ (n) +
1

24
σ (n)−

1
4

nσ
�

n
10

�

+
1
24
σ

�

n
10

�

+
1

312
σ3 (n) +

1
78
σ3

�

n
2

�

+
25
312

σ3

�

n
5

�

+
25
78
σ3

�

n
10

�

−
1

120
τ4,10(n)−

3
260

τ4,5(n)−
3
65
τ4,5(

n
2
).

Similarly,

E2 (2z) E2 (5z) =1+
+∞
∑

n=1

(−24
�

σ(
n
5
) +σ

�

n
2

��

+ 576W2,5(n))q
n

=
1

130
E4 +

2
65

E4,2 +
5

26
E4,5 +

10
13
∆4,5 (z) +

24
5
∆4,10 (z)−

432
65
∆4,5 (z)

−
1728
65
∆4,5 (2z) +

12
5

DΦ1,5 +
3
5
Φ1,2 +

6
5

DE2.



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 91

So,

W2,5(n) =−
1
20

nσ(
n
2
) +

1
24
σ(

n
2
)−

1
8

nσ(
n
5
) +

1
24
σ(

n
5
) +

1
312

σ3(n)

+
1
78
σ3(

n
2
) +

25
312

σ3(
n
5
) +

25
78
σ3(

n
10
) +

1
120

τ4,10 (n)

−
3

260
τ4,5 (n)−

3
65
τ4,5

�

n
2

�

.

2.5. Evaluation of W55(n) W5,11(n)

The vector space M4[Γ0(55)] has dimension 20. There are 4 linearly independent Eisen-
stein forms E4, E4,5, E4,11, E4,55, 6 old cusp forms

∆4,5,∆4,5 (11z) ,∆4,11,2 (z)

(the Galois conjugate of

∆4,11,1 (z) = q+ sq2 + · · ·+ (−88s+ 77)q55 +O(q56)

with polynomial x2 − 2x − 2, s = 1+
p

3),

∆4,11,1 (5z) ,∆4,11,2 (5z)

and 10 newforms

∆4,55,1 = q+ q2 + · · ·+ (−55)q55 +O(q56),

∆4,55,2,∆4,55,3,∆4,55,4,∆4,55,5,∆4,55,6,

∆4,55,7,∆4,55,8,∆4,55,9,∆4,55,10.

Here

∆4,55,2 = q+ rq2 + · · ·+ (−55)q55 +O(q56), r =
−7+

p
17

2
,

and ∆4,55,3 is the Galois conjugate of ∆4,55,2 by x2 + 7x + 8. ∆4,55,5, ∆4,55,6, are the Galois
conjugates of

∆4,55,4 = q+ uq2 + · · ·+ 55q55 +O(q56)

by x3 − 5x2 − 11x + 59, the roots are u, p, 5− u− p.

∆4,55,8,∆4,55,9,∆4,55,10

are the Galois conjugates of

∆4,55,7 = q+wq2 + · · ·+ 55q55 +O(q56)

by
x4 − x3 − 25x2 + 9x + 96.



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 92

The roots are w, g, h and 1−w− g − h.
The vector space M2[Γ0(55)] has dimension 8 and is spanned by three new forms ∆2,55,1,

∆2,55,2, ∆2,55,3 (z) (the Galois conjugate of ∆2,55,2 (z) by x2 − 2x − 1, t = 1+
p

2). Moreover,
there are two old forms

∆2,11 = q− 2q2 + · · ·+ q55 +O
�

q56
�

(the lifting of unique newform in S2[Γ0(11)]),∆2,11 (5z) and three Eisenstein formsΦ1,5,Φ1,11,
Φ1,55.

Consequently, we get

E2 (z) E2 (55z) =1+
+∞
∑

n=1

(−24
�

σ(n) +σ
�

n
55

��

+ 576
+∞
∑

n=1

W55(n))q
n

=
1

3172
E4 (z) +

25
3172

E4,5 (z) +
121
3172

E4,11 (z) +
3025
3172

E4,55 (z) +
324
55

DΦ1,55

+
12
55

DE2 −
864
2665

∆4,5 (z)−
104544
2665

∆4,5 (11z)

+ (
243072
515999

s−
1074240
515999

)∆4,11,1 (z) + (
6076800
515999

s−
26856000
515999

)∆4,11,1 (5z)

− (
243072
515999

s+
588096
515999

)∆4,11,2 (z)− (
6076800
515999

s+
14702400
515999

)∆4,11,2(5z)

+ (
275220
2238559

u2 −
1271088
2238559

u−
7581348
2238559

)∆4,55,4 (z)

+ ((
−275220
2238559

u+
105012

2238559
)p−

275220
2238559

u2 +
1376100
2238559

u−
4553928
2238559

)∆4,55,5 (z)

+ ((
275220
2238559

u−
105012

2238559
)p−

105012
2238559

u−
4028868
2238559

)∆4,55,6 (z)

+ (
67959

1411190
w3 +

17223
705595

w2 +
1963209
1411190

w+
2276304
705595

)∆4,55,7 (z)

+ ((
−67959
1411190

w+
20481
282238

)g2 + (−
67959

1411190
w2 +

67959
1411190

w−
132117
705595

)g

−
67959

1411190
w3 +

67959
1411190

w2 +
339795
282238

w−
5164239
1411190

)∆4,55,8 (z)

+ (((
67959

1411190
w−

20481
282238

)g + (−
20481
282238

w−
161829

1411190
))h+ (

67959
1411190

w−
20481
282238

)g2

+ (
67959

1411190
w2 −

85182
705595

w+
20481
282238

)g −
20481
282238

w2 +
20481

282238
w−

1302057
705595

)∆4,55,9 (z)

+ (((
−67959
1411190

w+
20481
282238

)g + (
20481

282238
w+

161829
1411190

))h

+ (
20481
282238

w+
161829

1411190
)g +

161829
1411190

w−
2765943
1411190

)∆4,55,10 (z) .

So,

W55 (n) =−
1

220
nσ (n) +

1
24
σ (n)−

1
4

nσ
�

n
55

�

+
1

24
σ

�

n
55

�



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 93

+
5

38064
σ3 (n) +

125
38064

σ3

�

n
5

�

+
605

38064
σ3

�

n
11

�

+
15125
38064

σ3

�

n
55

�

+ (
7645

35816944
u2 −

8827
8954236

u−
210593

35816944
)τ4,55,4 (n)

+ ((−
7645

35816944
u+

2917
35816944

)p+ (−
7645

35816944
u2 +

38225
35816944

u−
63249

17908472
))τ4,55,5 (n)

+ ((
7645

35816944
u−

2917
35816944

)p+ (−
2917

35816944
u−

111913
35816944

))τ4,55,6 (n)

+ (
7551

90316160
w3 +

5741
135474240

w2 −
654403

270948480
w−

47423
8467140

)τ4,55,7 (n)

+ ((−
7551

90316160
w+

6827
54189696

)g2 + (−
7551

90316160
w2 −

44039
135474240

)g

+ (−
7551

90316160
w3 +

7551
90316160

w2 +
37755

18063232
w−

1721413
270948480

))τ4,55,8 (n)

+ (((
7551

90316160
w−

6827
54189696

)g + (−
6827

54189696
w−

17981
90316160

))h

+ ((
7551

90316160
w−

6827
54189696

)g2 + (
7551

90316160
w2 −

14197
67737120

w+
6827

54189696
)g

+ (−
6827

54189696
w2 +

6827
54189696

6w−
144673

45158080
)))τ4,55,9 (n)

+ (((−
7551

90316160
w+

6827
54189696

)g + (
6827

54189696
w+

17981
90316160

))h

+ ((
6827

54189696
w+

17981
90316160

)g + (
17981

90316160
w−

307327
90316160

)))τ4,55,10 (n)

+ (
422

515999
s−

1865
515999

)τ4,11,1 (n) + (
10550

515999
s−

46625
515999

)τ4,11,1

�

n
5

�

+ (−
422

515999
s−

1021
515999

)τ4,11,2 (n) + (−
10550
515999

s−
25525
515999

)τ4,11,2

�

n
5

�

−
3

5330
τ4,5 (n)−

363
5330

τ4,5

�

n
11

�

.

Similarly,

E2 (5z) E2 (11z) =1+
+∞
∑

n=1

�

−24
�

σ(
n

11
) +σ

�

n
5

��

+ 576W5,11(n)
�

qn

=
1

3172
E4 (z) +

25
3172

E4,5 (z) +
121
3172

E4,11 (z) +
3025
3172

E4,55 (z)

+
24
55

DΦ1,5 +
12
11

DΦ1,11 +
12
55

DE2 −
864

2665
∆4,5 (z)−

104544
2665

∆4,5 (11z)

+ (
243072
515999

s−
1074240
515999

)∆4,11,1 (z) + (
6076800
515999

s−
26856000
515999

)∆4,11,1 (5z)

− (
243072
515999

s+
588096
515999

)∆4,11,2 (z)− (
6076800
515999

s+
14702400
515999

)∆4,11,2(5z)



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 94

+ (
275220
2238559

u2 +
1271088
2238559

u−
7581348
2238559

)∆4,55,4 (z)

+ ((−
275220
2238559

u+
105012
2238559

)p−
275220
2238559

u2 +
1376100
2238559

u−
4553928
2238559

)∆4,55,5 (z)

+ ((
275220

2238559
u−

105012
2238559

)p−
105012
2238559

u−
4028868
2238559

)∆4,55,6 (z)

+ (−
67959

1411190
w3 −

17223
705595

w2 +
1963209
1411190

w+
2276304
705595

)∆4,55,7 (z)

+ ((
67959

1411190
w−

20481
282238

)g2 + (
67959

1411190
w2 −

67959
1411190

w+
132117
705595

)g

+
67959

1411190
w3 −

67959
1411190

w2 −
339795
282238

w+
5164239
1411190

)∆4,55,8 (z)

+ (((−
67959

1411190
w+

20481
282238

)g + (
20481
282238

w+
161829

1411190
))h

− (
67959

1411190
w−

20481
282238

)g2 − (
67959

1411190
w2 −

85182
705595

w+
20481
282238

)g

+
20481
282238

w2 −
20481
282238

w+
1302057
705595

)∆4,55,9 (z)

+ (((
67959

1411190
w−

20481
282238

)g − (
20481
282238

w+
161829
1411190

))h

− (
20481
282238

w+
161829
1411190

)g −
161829
1411190

w+
2765943
1411190

)∆4,55,10 (z) .

So,

W5,11 (n) =−
1
44

nσ
�

n
5

�

+
1

24
σ

�

n
5

�

−
1

20
nσ
�

n
11

�

+
1
24
σ

�

n
11

�

+
5

38064
σ3 (n) +

125
38064

σ3

�

n
5

�

+
605

38064
σ3

�

n
11

�

+
15125
38064

σ3

�

n
55

�

+(
7645

35816944
u2 −

8827
8954236

u−
210593

35816944
)τ4,55,4 (n)

+ ((−
7645

35816944
u+

2917
35816944

)p+ (−
7645

35816944
u2 +

38225
35816944

u−
63249

17908472
))τ4,55,5 (n)

+ ((
7645

35816944
u−

2917
35816944

)p+ (−
2917

35816944
u−

111913
35816944

))τ4,55,6 (n)

+ (−
7551

90316160
w3 −

5741
135474240

w2 +
654403

270948480
w+

47423
8467140

)τ4,55,7 (n)

+ ((
7551

90316160
w−

6827
54189696

)g2 + (
7551

90316160
w2 −

7551
90316160

w+
44039

135474240
)g

+ (
7551

90316160
w3 −

7551
90316160

w2 −
37755

18063232
w+

1721413
270948480

))τ4,55,8 (n)

+ (((−
7551

90316160
w+

6827
54189696

)g + (
6827

54189696
w+

17981
90316160

))h



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 95

+ ((−
7551

90316160
w+

6827
54189696

)g2 + (−
7551

90316160
w2 +

14197
67737120

w−
6827

54189696
)g

+ (
6827

54189696
w2 +

6827
54189696

w+
144673

45158080
)))τ4,55,9 (n)

+ (((
7551

90316160
w−

6827
54189696

)g + (−
6827

54189696
w−

17981
90316160

))h

+ ((−
6827

54189696
w−

17981
90316160

)g + (−
17981

90316160
w+

307327
90316160

)))τ4,55,10 (n)

+ (
422

515999
s−

1865
515999

)τ4,11,1 (n)

+ (
10550

515999
s−

46625
515999

)τ4,11,1

�

n
5

�

+ (−
422

515999
s−

1021
515999

)τ4,11,2 (n)

+ (−
10550
515999

+ s
25525

515999
)τ4,11,2

�

n
5

�

−
3

5330
τ4,5 (n)−

363
5330

τ4,5

�

n
11

�

.

By the same method, we calculate

Wσa,τb (n) =
∑

k,m∈Z
ak+bm=n

σ (k)τ2,11 (m) for some a and b.

Wσ,τ5 (n) =−
1
20
τ4,55,1 (n) + (−

15
2006

r −
275

4012
)τ4,55,2 (n) + (

15
2006

r −
65

4012
)τ4,55,3 (n)

+ (
1695

4477118
u2 −

73771
17908472

u+
253919

17908472
)τ4,55,4 (n)

+ ((−
1695

4477118
u−

39871
17908472

)p+ (−
1695

4477118
u2 +

8475
4477118

u+
328499

17908472
))τ4,55,5 (n)

+ ((
1695

4477118
u+

39871
17908472

)p+ (
39871

17908472
u+

16143
2238559

))τ4,55,6 (n)

+ (
1513

2463168
w3 −

5999
3078960

w2 −
162281

12315840
w+

11703
256580

)τ4,55,7 (n)

+ ((−
1513

2463168
w−

5477
4105280

)g2 + (−
1513

2463168
w2 +

1513
2463168

w+
2237

1026320
)g

+ (−
1513

2463168
w3 +

1513
2463168

w2 +
37825

2463168
w+

164553
4105280

))τ4,55,8 (n)

+ (((
1513

2463168
w+

5477
4105280

)g + (
5477

4105280
w+

3471
4105280

))h

+ ((
1513

2463168
w+

5477
4105280

)g2 + (
1513

2463168
w2 +

4433
6157920

w−
5477

4105280
)g

+ (
5477

4105280
w2 −

5477
4105280

w+
6907

1026320
)))τ4,55,9 (n)

+ (((−
1513

2463168
w−

5477
4105280

)g + (−
5477

4105280
w−

3471
4105280

))h

+ ((−
5477

4105280
w−

3471
4105280

)g + (−
3471

4105280
w+

31099
4105280

)))τ4,55,10 (n)



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 96

+ (−
691

90742
s+

675
45371

)τ4,11,1 (n) + (−
318475
2177808

s+
885625

1088904
)τ4,11,1

�

n
5

�

+ (
691

90742
s−

16
45371

)τ4,11,2 (n) + (
318475
2177808

s+
94525
181484

)τ4,11,2

�

n
5

�

−
4

205
τ4,5 (n) +

3993
410

τ4,5

�

n
11

�

−
1
4

nτ2,11

�

n
5

�

+
1
24
τ2,11

�

n
5

�

,

Wσ5,τ (n) =−
1
20
τ4,55,1 (n) + (

15
2006

r +
275
4012

)τ4,55,2 (n) + (−
15

2006
r +

65
4012

)τ4,55,3 (n)

+ (
1695

4477118
u2 −

73771
17908472

u+
253919

17908472
)τ4,55,4 (n)

+ ((−
1695

4477118
u−

39871
17908472

)p+ (−
1695

4477118
u2 +

8475
4477118

u+
328499

17908472
))τ4,55,5 (n)

+ ((
1695

4477118
u+

39871
17908472

)p+ (
39871

17908472
u+

16143
2238559

))τ4,55,6 (n)

+ (−
1513

2463168
w3 +

5999
3078960

w2 +
162281

12315840
w−

11703
256580

)τ4,55,7 (n)

+ ((
1513

2463168
w+

5477
4105280

)g2 + (
1513

2463168
w2 −

1513
2463168

w−
2237

1026320
)g

+ (
1513

2463168
w3 −

1513
2463168

w2 −
37825

2463168
w−

164553
4105280

))τ4,55,8 (n)

+ (((−
1513

2463168
w−

5477
4105280

)g + (−
5477

4105280
w−

3471
4105280

))h

+ ((−
1513

2463168
w−

5477
4105280

)g2 + (−
1513

2463168
w2 −

4433
6157920

w+
5477

4105280
)g

+ (−
5477

4105280
w2 +

5477
4105280

w−
6907

1026320
)))τ4,55,9 (n)

+ (((
1513

2463168
w+

5477
4105280

)g + (
5477

4105280
w+

3471
4105280

))h

+ ((
5477

4105280
w+

3471
4105280

)g + (
3471

4105280
w−

31099
4105280

)))τ4,55,10 (n)

+ (−
12739

2177808
s+

35425
1088904

)τ4,11,1 (n) + (−
17275
90742

s+
16875
45371

)τ4,11,1

�

n
5

�

+ (
12739

2177808
s+

3781
181484

)τ4,11,2 (n) + (
17275
90742

s−
400

45371
)τ4,11,2

�

n
5

�

−
4

205
τ4,5 (n) +

3993
410

τ4,5

�

n
11

�

−
1
20

nτ2,11 (n) +
1
24
τ2,11 (n)

Wσ55,τ (n) =−
1

220
τ4,55,1 (n) + (−

15
22066

r −
25

4012
)τ4,55,2 (n) + (

15
22066

r −
65

44132
)τ4,55,3 (n)

+ (−
1695

49248298
u2 +

73771
196993192

u−
253919

196993192
)τ4,55,4 (n)



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 97

+ ((
1695

49248298
u+

39871
196993192

)p+ (
1695

49248298
u2 −

8475
49248298

u−
328499

196993192
))τ4,55,5 (n)

+ ((−
1695

49248298
u−

39871
196993192

)p+ (−
39871

196993192
u−

16143
24624149

))τ4,55,6 (n)

+ (−
1513

27094848
w3 +

5999
33868560

w2 +
162281

135474240
w−

11703
2822380

)τ4,55,7 (n)

+ ((
1513

27094848
w+

5477
45158080

)g2 + (
1513

27094848
w2 −

1513
27094848

w−
2237

11289520
)g

+ (
1513

27094848
w3 −

1513
27094848

w2 −
37825

27094848
w−

164553
45158080

))τ4,55,8 (n)

+ (((−
1513

27094848
w−

5477
45158080

)g

+ (−
5477

45158080
w−

3471
45158080

))h+ ((−
1513

27094848
w−

5477
45158080

)g2

+ (−
1513

27094848
w2 −

403
6157920

w+
5477

45158080
)g

+ (−
5477

45158080
w2 +

5477
45158080

w−
6907

11289520
)))τ4,55,9 (n)

+ (((
1513

27094848
w+

5477
45158080

)g + (
5477

45158080
w+

3471
45158080

))h

+ ((
5477

45158080
w+

3471
45158080

)g + (
3471

45158080
w−

31099
45158080

)))τ4,55,10 (n)

+ (
12739

23955888
s−

35425
11977944

)τ4,11,1 (n) + (
17275

998162
s−

16875
499081

)τ4,11,1

�

n
5

�

+ (−
12739

23955888
s−

3781
1996324

)τ4,11,2 (n) + (−
17275
998162

s+
400

499081
)τ4,11,2

�

n
5

�

−
3

410
τ4,5 (n) +

44
205

τ4,5

�

n
11

�

−
1

220
nτ2,11 (n) +

1
24
τ2,11 (n) ,

Wσ11,τ5 (n) =−
1

220
τ4,55,1 (n) + (

15
22066

r +
25

4012
)τ4,55,2 (n) + (−

15
22066

r +
65

44132
)τ4,55,3 (n)

+ (−
1695

49248298
u2 +

73771
196993192

u−
253919

196993192
)τ4,55,4 (n)

+ ((
1695

49248298
u+

39871
196993192

)p+ (
1695

49248298
u2 −

8475
49248298

u−
328499

196993192
))τ4,55,5 (n)

+ ((−
1695

49248298
u−

39871
196993192

)p+ (−
39871

196993192
u−

16143
24624149

))τ4,55,6 (n)

+ (
1513

27094848
w3 −

5999
33868560

w2 −
162281

135474240
w+

11703
2822380

)τ4,55,7 (n)

+ ((−
1513

27094848
w−

5477
45158080

)g2 + (−
1513

27094848
w2 +

1513
27094848

w+
2237

11289520
)g

+ (−
1513

27094848
w3 +

1513
27094848

w2 +
37825

27094848
w+

164553
45158080

))τ4,55,8 (n)



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 98

+ (((
1513

27094848
w+

5477
45158080

)g + (
5477

45158080
w+

3471
45158080

))h

+ ((
1513

27094848
w+

5477
45158080

)g2 + (
1513

27094848
w2 +

403
6157920

w−
5477

45158080
)g

+ (
5477

45158080
w2 −

5477
45158080

w+
6907

11289520
)))τ4,55,9 (n)

+ (((−
1513

27094848
w−

5477
45158080

)g + (−
5477

45158080
w−

3471
45158080

))h

+ ((−
5477

45158080
w−

3471
45158080

)g + (−
3471

45158080
w+

31099
45158080

)))τ4,55,10 (n)

+ (
691

998162
s−

675
499081

)τ4,11,1 (n) + (
318475

23955888
s−

885625
11977944

)τ4,11,1

�

n
5

�

+ (−
691

998162
s+

16
499081

)τ4,11,2 (n) + (−
318475

23955888
s−

94525
1996324

)τ4,11,2

�

n
5

�

−
3

410
τ4,5 (n) +

44
205

τ4,5

�

n
11

�

−
1

44
nτ2,11

�

n
5

�

+
1
24
τ2,11

�

n
5

�

,

Wτ,τ5 (n) =(−
445

54599
u2 +

1329
218396

u+
26429
218396

)τ4,55,4 (n)

+ ((
445

54599
u−

7571
218396

)p+ (
445

54599
u2 −

2225
54599

u+
6849

218396
))τ4,55,5 (n)

+ ((−
445

54599
u+

7571
218396

)p+ (
7571

218396
u−

15503
109198

))τ4,55,6 (n)

+ (
31

4614
s−

55
4614

)τ4,11,1 (n) + (
775
4614

s−
1375
4614

)τ4,11,1

�

n
5

�

+ (−
31

4614
s+

7
4614

)τ4,11,2 (n) + (−
775

4614
s+

175
4614

)τ4,11,2

�

n
5

�

.

3. Application to the Number of Representations

Theorem 1. The number Ni (n) of representations of a positive integer n by the quadratic forms

Q1 =x2
1 + x1 x2 + x2

2 + x2
3 + x3 x4 + x2

4 + 9
�

x2
5 + x5 x6 + x2

6 + x2
7 + x7 x8 + x2

8

�

,

Q2 =x2
1 + 2x2

2 + x2
3 + 2x2

4 + 5
�

x2
5 + 2x2

6 + x2
7 + 2x2

8

�

,

Q3 =x2
1 + x1 x2 + 3x2

2 + x2
3 + x3 x4 + 3x2

4 + 5
�

x2
5 + x5 x6 + 3x2

6 + x2
7 + x7 x8 + 3x2

8

�

,

are equal to

N1 (n) =
4
9
σ3(n)−

76
9
σ3(

n
3
)− 76σ3(

n
9
) + 324σ3(

n
27
) +

16
3
τ4,27,1 (n)

+
1
9
(4w+ 24)τ4,27,3 (n) +

1
9
(−4w+ 24)τ4,27,4 (n) +

8
9
τ4,9 (n) + 8τ4,27,1

�

n
3

�



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 99

N2 (n) =
24
5

nσ
�

n
8

�

− 32σ
�

n
8

�

+ 24nσ
�

n
40

�

− 32σ
�

n
40

�

+
2

13
σ3 (n)−

2
13
σ3

�

n
2

�

−
8
13
σ3

�

n
4

�

+
50
13
σ3

�

n
5

�

−
32
13
σ3

�

n
8

�

−
50
13
σ3

�

n
10

�

−
200
13
σ3

�

n
20

�

−
800
13
σ3

�

n
40

�

+
8
7
τ4,40,3 (n) +

6
7
τ4,8 (n) +

150
7
τ4,8

�

n
5

�

+
24
13
τ4,5 (n) +

184
13
τ4,5

�

n
2

�

+
736
13
τ4,5

�

n
4

�

+
8064
65

τ4,5

�

n
8

�

,

N3(n) =
60
793

σ3 (n) +
1500
793

σ3

�

n
5

�

+
7260
793

σ3

�

n
11

�

+
181500

793
σ3

�

n
55

�

+ (−
94238

2238559
u2 +

172872
2238559

u+
2848918
2238559

)τ4,55,4 (n)

+ ((
94238

2238559
u−

298318
2238559

)p+ (
94238

2238559
u2 −

471190
2238559

u+
1812300
2238559

))τ4,55,5 (n)

+ ((−
94238

2238559
u+

298318
2238559

)p+ (
298318
2238559

u+
320710
2238559

))τ4,55,6 (n) + (−
26716
140727

s

+
111400
140727

)τ4,11,1 (n) + (−
667900
140727

s+
2785000
140727

)τ4,11,1

�

n
5

�

+ (
26716/140727

140727
s+

57968
140727

)τ4,11,2 (n) + (
667900
140727

s+
1449200
140727

)τ4,11,2

�

n
5

�

+
264
533

τ4,5 (n) +
31944

533
τ4,5

�

n
11

�

respectively.

Proof. Let

r1 (l) = #

� �

x1, x2, x3, x4

�

∈ Z4 :
x2

1 + x1 x2 + x2
2 + x2

3 + x3 x4 + x2
4 = l

�

for l ∈ {0} ∪N. Since M2(Γ0 (3)) is spanned by Φ1,3, it follows that

r1 (l) = 12
�

σ (l)− 3σ
�

l
3

��

.

Now

N1 (n) =r1 (0) r1

�

n
9

�

+ r1 (n) r1 (0) +
∑

l,m∈N
l+9m=n

r1 (l) r1 (m)

=12σ
�

n
9

�

− 36σ
�

n
27

�

+ 12σ (n)− 36σ
�

n
3

�



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 100

+
∑

l,m∈N
l+9m=n

�

12σ (l)− 36σ
�

l
3

���

12σ (m)− 36σ
�

m
3

��

=12σ
�

n
9

�

− 36σ
�

n
27

�

+ 12σ (n)− 36σ
�

n
3

�

+ 144
∑

l,m∈N
l+9m=n

σ (l)σ (m)− 432
∑

l,m∈N
l+9m=n

σ (l)σ
�

m
3

�

− 432
∑

l,m∈N
l+9m=n

σ

�

l
3

�

σ (m) + 1296
∑

l,m∈N
l+9m=n

σ

�

l
3

�

σ

�

m
3

�

=12σ
�

n
9

�

− 36σ
�

n
27

�

+ 12σ (n)− 36σ
�

n
3

�

+ 144W9 (n)− 432W27 (n)− 432W3,9 (n) + 1296W9

�

n
3

�

=12σ
�

n
9

�

− 36σ
�

n
27

�

+ 12σ (n)− 36σ
�

n
3

�

+
2
3
σ3(n) +

16
3
σ3

�

n
3

�

+ 54σ3

�

n
9

�

.

Let
r2 (l) = #

��

x1, x2, x3, x4

�

∈ Z4 : x2
1 + 2x2

2 + x2
3 + 2x2

4 = l
	

for l ∈ {0} ∪N.
M2

�

Γ0 (8)
�

is spanned by three linearly independent Eisenstein series

Φ1,2,Φ1,4 and Φ1,8.

The theta series of x2
1 + 2x2

2 + x2
3 + 2x2

4 is

1+4q+ 8q2 + 16q3 + 24q4 + 24q5 + . . .

=
1
12
(1+

+∞
∑

n=1

�

24σ (n)− 48σ
�

n
2

��

qn)

−
1
4
(1+

+∞
∑

n=1

�

8σ (n)− 32σ
�

n
4

��

qn)

+
7
6
(1+

+∞
∑

n=1

�

24
7
σ (n)−

8 · 24
7
·σ
�

n
8

��

qn),

so,

r2 (l) = 4σ (l)− 4σ
�

l
2

�

+ 8σ
�

l
4

�

− 32σ
�

l
8

�

.

Now

N2 (n) =r2 (0) r2

�

n
5

�

+ r2 (n) r2 (0) +
∑

l,m∈N
l+5m=n

r2 (l) r2 (m)



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 101

=4σ
�

n
5

�

− 4σ
�

n
10

�

+ 8σ
�

n
20

�

− 32σ
�

n
40

�

+ 4σ (n)− 4σ
�

n
2

�

+ 8σ
�

n
4

�

− 32σ
�

n
8

�

+

∑

l,m∈N
l+5m=n

�

4σ (l)− 4σ
�

l
2

�

+ 8σ
�

l
4

�

− 32σ
�

l
8

��

�

4σ (m)− 4σ
�

m
2

�

+ 8σ
�

m
4

�

− 32σ
�

m
8

��

.

So,

N2 (n) =4σ (n)− 4σ
�

n
2

�

+ 8σ
�

n
4

�

+ 4σ
�

n
5

�

− 32σ
�

n
8

�

− 4σ
�

n
10

�

+ 8σ
�

n
20

�

− 32σ
�

n
40

�

+ 16W5 (n)

− 16W10 (n) + 32W20 (n)− 128W40 (n)− 16W2,5 (n)

+ 16W5

�

n
2

�

− 32W10

�

n
2

�

+ 128W20

�

n
2

�

+ 32W4,5 (n)− 32W2,5

�

n
2

�

+ 64W5

�

n
4

�

− 32 · 8W10

�

n
4

�

− 32 · 4W5,8 (n) + 32 · 4W4,5

�

n
2

�

− 32 · 8W2,5

�

n
4

�

+ 32 · 8W5

�

n
8

�

.

Let

r3 (l) = #

� �

x1, x2, x3, x4

�

∈ Z4 :
x2

1 + x1 x2 + 3x2
2 + x2

3 + x3 x4 + 3x2
4 = l

�

for l ∈ {0} ∪N. Since M2

�

Γ0 (11)
�

is spanned by Φ1,11 and

∆2,11(z) =
∞
∑

n=1

τ2,11 (n)q
n =

�

∆
�

q
�

∆
�

q11
��1/12

=q
∞
∏

n=1

�

1− qn
�2
∞
∏

n=1

�

1− q11n
�2

,

we have

r3 (l) =
12
5
σ (l)−

132
5
σ

�

l
11

�

+
8
5
τ2,11 (l) .

So,

N3(n) =r3 (0) r3

�

n
5

�

+ r3 (n) r3 (0) +
∑

l,m∈N
l+5m=n

r3 (l) r3 (m)



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 102

=
12
5
σ (n)−

132
5
σ

�

n
11

�

+
8
5
τ2,11 (n) +

12
5
σ

�

n
5

�

−
132
5
σ

�

n
55

�

+
8
5
τ2,11

�

n
5

�

+
144
25

W5 (n)

−
1584

25
W55 (n)−

1584
25

W5,11 (n) +
17424

25
W5(

n
11
)

+
96
25
(Wσ,τ5 +Wσ5,τ)−

1056
25
(Wσ11,τ5 +Wσ55,τ) +

64
25

Wτ,τ5

where

Wσ,τ5 =
∑

l,m∈N
l+5m=n

σ (l)τ2,11 (m) , Wσ5,τ =
∑

l,m∈N
l+5m=n

τ2,11 (l)σ (m)

Wσ11,τ5 =
∑

l,m∈N
11l+5m=n

σ (l)τ2,11 (m) , Wσ55,τ =
∑

l,m∈N
l+55m=n

τ2,11 (l)σ (m)

Wτ,τ5 =
∑

l,m∈N
l+5m=n

τ2,11 (l)τ2,11 (m) .

3.1. Numerical Examples

In Tables 1 through 6, we show the calculations which have been done by the formulas
obtained in this article.



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 103

Table 1: Some Convolution Sums and the Representation Numbers N1, N2 for n= 1− 25.

n W27(n) W3,9(n) N1(n) W40(n) W5,8(n) N2 (n)
1 0 0 12 0 0 4
2 0 0 36 0 0 8
3 0 0 12 0 0 16
4 0 0 84 0 0 24
5 0 0 72 0 0 28
6 0 0 36 0 0 48
7 0 0 96 0 0 64
8 0 0 180 0 0 88
9 0 0 24 0 0 148
10 0 0 360 0 0 152
11 0 0 576 0 0 208
12 0 1 228 0 0 288
13 0 0 1176 0 1 280
14 0 0 1152 0 0 464
15 0 3 504 0 0 496
16 0 0 1524 0 0 536
17 0 0 2376 0 0 840
18 0 4 216 0 3 776
19 0 0 3264 0 0 1136
20 0 0 3528 0 0 1320
21 0 10 1536 0 3 1216
22 0 0 5472 0 0 1856
23 0 0 6336 0 4 1728
24 0 15 2340 0 0 2336
25 0 0 8292 0 0 2836



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 104

Table 2: The Same Convolution Sums and the Representation Numbers N1, N2 for n= 26− 50.

n W27(n) W3,9(n) N1(n) W40(n) W5,8(n) N2 (n)
26 0 0 9576 0 9 2448
27 0 24 888 0 0 3616
28 1 0 11472 0 7 3264
29 3 0 12024 0 4 3992
30 4 33 4536 0 0 5136
31 7 0 12624 0 12 4032
32 6 0 15444 0 0 5912
33 12 45 5328 0 6 5824
34 8 0 19656 0 12 6224
35 15 0 16560 0 0 8512
36 13 65 1752 0 21 6648
37 18 0 20760 0 7 8376
38 12 0 27936 0 12 9120
39 28 77 8232 0 16 9184
40 14 0 30312 0 0 12520
41 24 0 29448 1 18 10536
42 24 102 12384 3 21 11936
43 31 0 34224 4 8 13168
44 18 0 43776 7 28 12768
45 39 143 3024 6 6 17164
46 20 0 52416 12 36 14448
47 42 0 42912 8 28 15520
48 32 155 19236 15 15 17824
49 36 0 52812 13 24 17764
50 24 0 67212 18 18 23432



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 105

Table 3: Some Convolution Sums for n= 1− 25.

n W4,10 (n) W2,20 (n) W20 W4,5 W2,10 W55(n) W5,11(n)
1 0 0 0 0 0 0 0
2 0 0 0 0 0 0 0
3 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0
7 0 0 0 0 0 0 0
8 0 0 0 0 0 0 0
9 0 0 0 1 0 0 0

10 0 0 0 0 0 0 0
11 0 0 0 0 0 0 0
12 0 0 0 0 1 0 0
13 0 0 0 3 0 0 0
14 1 0 0 3 3 0 0
15 0 0 0 0 0 0 0
16 0 0 0 0 4 0 1
17 0 0 0 4 0 0 0
18 3 0 0 9 7 0 0
19 0 0 0 4 0 0 0
20 0 0 0 0 6 0 0
21 0 0 1 7 0 0 3
22 4 0 3 12 15 0 0
23 0 1 4 12 0 0 0
24 3 0 7 7 17 0 0
25 0 3 6 6 0 0 0



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 106

Table 4: The Same Convolution Sums for n= 26− 50.

n W4,10 (n) W2,20 (n) W20 W4,5 W2,10 W55(n) W5,11(n)
26 7 0 12 21 27 0 4
27 0 4 8 16 0 0 3
28 9 0 15 21 34 0 0
29 0 7 13 18 0 0 0
30 6 0 18 18 36 0 0
31 0 6 12 28 0 0 7
32 12 0 28 28 52 0 9
33 0 12 14 28 0 0 0
34 16 0 24 26 64 0 0
35 0 8 24 48 0 0 0
36 21 0 31 24 75 0 6
37 0 15 18 49 0 0 12
38 20 0 39 39 91 0 4
39 0 13 20 60 0 0 0
40 18 0 42 56 102 0 0
41 0 18 35 42 0 0 12
42 31 0 45 55 122 0 21
43 0 15 36 93 0 0 12
44 43 0 81 56 155 0 0
45 0 37 49 99 0 0 0
46 41 0 78 54 169 0 8
47 0 26 64 123 0 0 18
48 45 0 101 92 193 0 16
49 0 45 69 101 0 0 7
50 42 0 126 77 228 0 0



B. Kendirli / Eur. J. Pure Appl. Math, 8 (2015), 81-110 107

Table 5: Some Convolution Sums and Representation Numbers N3 for n= 1− 25.

n Wσ,τ5(n) Wσ5,τ(n) Wσ55,τ(n) Wσ11,τ5(n) Wτ,τ5(n) N3(n)
1 0 0 0 0 0 4
2 0 0 0 0 0 4
3 0 0 0 0 0 8
4 0 0 0 0 0 20
5 0 0 0 0 0 20
6 1 1 0 0 1 48
7 3 -2 0 0 -2 32
8 4 -1 0 0 -1 68
9 7 2 0 0 2 108
10 6 1 0 0 1 108
11 10 5 0 0 0 148
12 2 -8 0 0 2 144
13 7 -3 0 0 2 216
14 -1 4 0 0 -6 256
15 6 1 0 0 -4 288
16 -13 11 0 1 -4 244
17 9 -16 0 0 4 392
18 -20 0 0 0 5 468
19 -9 6 0 0 6 576
20 -18 -3 0 0 2 636
21 -3 14 0 -2 -6 704
22 -40 -30 0 0 0 628
23 4 9 0 0 -6 1064
24 -27 18 0 0 0 1216
25 -12 -2 0 0 -6 1364



REFERENCES 108

Table 6: The Same Convolution Sums and Representation Numbers N3 for n= 26− 50.

n Wσ,τ5(n) Wσ5,τ(n) Wσ55,τ(n) Wσ11,τ5(n) Wτ,τ5(n) N3(n)
26 -17 14 0 -1 -2 1416
27 -9 -42 0 3 8 1176
28 -34 21 0 0 14 1920
29 29 14 0 0 -2 2232
30 -35 -10 0 0 -14 2400
31 -15 13 0 2 -6 2024
32 20 -59 0 -6 -10 2580
33 20 25 0 0 -2 3176
34 -35 40 0 0 1 3752
35 46 -9 0 0 0 4208
36 12 1 0 1 10 4068
37 21 -51 0 -3 6 3600
38 26 27 0 4 16 5056
39 10 20 0 0 -6 5312
40 5 -20 0 0 22 6460
41 80 18 0 2 0 5464
42 2 -49 0 6 -5 5264
43 -7 31 0 -8 -12 6592
44 80 70 0 0 -24 8516
45 38 -27 0 0 16 8964
46 -20 -33 0 -2 0 8624
47 47 -36 0 3 -2 7856
48 54 23 0 -4 -33 10144
49 16 34 0 7 -6 9684
50 49 -21 0 0 -12 13492

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