THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47. ˙ I˙ BARIS ¸ KENDIRL

Abstract. In this study bases of S5 (Γ0 (47)),S6 (Γ0 (47)) are determined and explicit formulas obtained for the number of representations of positive integers by all possible direct sums (48 different combinations) of the quadratic forms from the class group of equivalence classes of quadratic forms with discriminant -47 whose representatives are x21 + x1 x2 + 12x22 , 3x21 ± x1 x2 + 4x22 , 2x21 ± x1 x2 + 6x22 .

1.

Introduction

This article is about the representation numbers of several direct sums quadratic forms √ of
discriminant -47.It is an extension of the work [3] The class number of Q −47 is 5 and the following quadratic forms F1 = x21 + x1 x2 + 12x22 , G1 = 3x21 + x1 x2 + 4x22 , H1 = 2x21 + x1 x2 + 6x22 , H10 = 2x21 − x1 x2 + 6x22 , G01 = 3x21 − x1 x2 + 4x22 , are the representatives. Its group structure is as follows: G1 , G21 = H1 , G31 = H10 , G41 = G01 , G51 = F1 . For the quadratic form F1 = x21 + x1 x2 + 12x22 , 2F1 = 2x21 + 2x1 x2 + 24x22 = (x1 , x2 )



2 1

  1 x1 , 24 x2

the determinant and a cofactor are D = 47, A22 = 2. So δ =

gcd( A2ii , Aij , f or1

≤ i ≤ j ≤ 2) = 1, N = 2/2

∆ = (−1)

D δ

= 47 and the discriminant is

47 = −47.

The character of F1 is the Kronecker symbol   −47 × χ (d) = for d ∈ (Z/47Z) . d

(1.1)

Similarly, for the quadratic form G1 = 3x21 + x1 x2 + 4x22 ,    6 1 x1 2G1 = 6x21 + 2x1 x2 + 8x22 = (x1 , x2 ) , 1 8 x2 Key words and phrases. Quadratic Forms, Representation Numbers, Theta Series Cusp Forms 11E25, 11E76. 1

2

˙ BARIS ¸ KENDIRL I˙

the determinant and a cofactor are D = 47, A21 = −1. So 2/2

δ = 1, N = D = 47, ∆ = (−1) and the character of G1 is [1.1] . For the quadratic form H1 = 2x21 + x1 x2 + 6x22 ,

2H1 = 4x21 + 2x1 x2 + 12x22 = (x1 , x2 )

47 = −47



  1 x1 , 12 x2

4 1

the determinant and a cofactor are D = 47, A21 = −1. So 2/2

δ = 1, N = D = 47, ∆ = (−1) and the character of H1 is [1.1] .

47 = −47

1.1. Weight 1 Case: The theta series of F1 , G1 , H1 are in M1 Γ0 (47) ,
√ if ρk is a character of the class group of Q −47 , i.e.,

−47 d





.Moreover,

2πimk/5 ρk (Gm for m = 0, 1, 2, 3, 4, k = 0, 1, 2, 3, 4, 1 )=e



. then the following set of Hecke eigenforms is a basis of M1 Γ0 (47) , −47 d   

−1  1 −1 fρk (q) = ΘF1 (q) + ρk ( G1 ) + ρk ( G1 ) Θ G1 (q) + ρk G21 + ρk G21 Θ 2 for k = 0, 1, 2. More precisely, the Eisenstein series 1 fρ0 (q) := (ΘF1 (q) + 2Θ G1 (q) + 2Θ H1 (q)) 2

is a basis of Eisenstein subspace E1 Γ0 (47) , −47 and since d 1 1√ 4π 1√ 1 2π =− + 5, cos =− 5− cos 5 4 4 5 4 4
the following two Gal Q/Q −conjugate newforms ! ( √ ! √ ! −1 − 5 1 −1 + 5 Θ H1 (q) , Θ G1 (q) + = fρ1 (q) = ΘF1 (q) + 2 2 2 !) √ ! √ ! −1 + 5 −1 − 5 1 Θ H1 (q) fρ2 (q) = Θ G1 (q) + ΘF1 (q) + 2 2 2

. is a basis of the cuspidal subspace S1 Γ0 (47) , −47 d Now, n th Fourier coefficient of fρ0 X  −47  = r (F1 , n) + 2r ( G1 , n) + 2r ( H1 , n) = 2 , d d|n

In particular, for n=p prime, we have  r (F1 , p) + 2r ( G1 , p) + 2r ( H1 , p) = 2 1 +



−47 p



  p  =2 1+ 47

H1

 (q)

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47.3

=

1 2

αn = n th Fourier coefficient of fρ1 ! √ ! √ ! −1 + 5 −1 − 5 r (F1 , n) + r ( G1 , n) + r ( H1 , n) . 2 2

αn = nth Fourier coefficient of fρ2 ! √ ! √ ! 1 −1 − 5 −1 + 5 = r (F1 , n) + r ( G1 , n) + r ( H1 , n) , 2 2 2 √
where αn is the conjugate in Q 5 . By solving these linear equations, we get the following formulas for the representation numbers of n by F1 , G1 , H1 in terms of the Fourier coefficients αn of fρ1 .   2 X −47 2 + (αn + αn ) r (F1 , n) = 3 d 3 d|n         X √ √ −47 1 10 −3 + 5  − αn  − 4 5 (αn − αn ) r ( G1 , n) = √ d 30 5 − 90 d|n √   1 X −47 αn 5 r ( H1 , n) = − − (αn − αn ) . 3 d 6 10 d|n

Since αnm αpn α47n

= αn · αm for gcd (n, m) = 1, and = αpn−1 αp − χ−47 (p) αpn−2 for prime p,p - 47, = αn47 for any positive integer n,

But αp ,for prime integers p, can be given as  1 if p=47, (since r (F1 , 47) = 2,
r (G1 , 47) = 0, r (H1 , 47) = 0)    p   
0 if 47 = −1  p  2√if  47 = 1
and F1 represents p . αp = p −1+ 5  if 47 = 1 and G1 represents p   2   √  
 p −1− 5  if 47 = 1 and H1 represents p 2 Here, we can also obtain the factorization of pO−47 by means of αp as

IIn fact, this ideal is

√

1 − if αp

=

47O−47 .So 47O−47

=

2 − if αp

=

3 − if αp

=

4 − if αp

=

5 − if αp

=


Moreover,η (z) η (47z) ∈ S1 Γ0 (47) , χ−47 .

1, then p = 47 so r (F1 , 47) = 2, i.e., there exists only one integr √ 2 47O−47 is ramified ideal. p 0, then = −1, so pO−47 remains prime in O−47 ,  47  p 2, then = 1, so r (F1 , p) = 4, so pO−47 splits into produ 47 √ ! p −1 + 5 , then = 1, and G1 represents p, so r (F1 , p) 2 47 √ ! p −1 − 5 , then = 1, and H1 represents p, so r (F1 , p 2 47

˙ BARIS ¸ KENDIRL I˙

4



√ √ Let Q −47 H be the Hilbert field of Q −47 ,i.e.,maximal abelian unramified




√ √ √ √ extension of Q −47 .Gal Q −47 H /Q −47 'class group of Q −47
√ hence it has 5 elements.Since a rational prime p splits in Q −47 if and only if √ 
√ −1± 5 p splits completely in Q −47 H .,i.e.,if and only if αp = 2,or αp = . 2

√ √ Q −47 H = F.Q −47 F should be an extension of Q of degree 5.What is F?The description of factorization of pOF is as follows: √ 2 1 − if αp = 1, then p = 47 since 47O−47 = 47O−47 is ramified ideal 47OF is also ramified? 2 − if αp 3 − if αp 4 − if αp 5 − if αp

p = −1, so pOF either remains prime in OF , or splits completely.  47  p = 2, then = 1, so r (F1 , p) = 4, so pOF splits completely in OF since each principal ideal in 47 √ ! p −1 + 5 , then = 1, and G1 represents p, so r (F1 , p) = 0, and so pOF splits complet = 2 47 √ ! p −1 − 5 = , then = 1, and H1 represents p, so r (F1 , p) = 0, and so pOF splits complet 2 47

=

0, then

1.2. Weight 4 Case. This has been covered in [1]

1.3. Weight 5 Case. By similar calculation as in [2], we get 47 dim M5 (Γ0 (47) , χ) = 3



1 1+ 47



1 + λ (1, 1, 47) − v−3 α (χ) − µ−3 β (χ) 2

2 1 − 0 + · 0 = 17 2 3   (−3 − 1) 47 1 1 dim S5 (Γ0 (N ) , χ) = − 1+ − λ (1, 1, 47) + v3 α (χ) + µ3 β (χ) 12 47 2 = 16 +

= 16 −

2 1 − 0 − · 0 = 15 2 3

There are 22 cases Q = F5 , G5 , H5 , F4 ⊕ G1 , F4 ⊕ H1 , F3 ⊕ G2 , F3 ⊕ H2 , F3 ⊕ G1 ⊕ H1 F2 ⊕ G 3 , F 2 ⊕ H 3 , F 2 ⊕ G 2 ⊕ H 1 , F 2 ⊕ H 2 ⊕ G 1 , F1 ⊕ G 4 , F 1 ⊕ H 4 , F 1 ⊕ G 3 ⊕ H 1 , F 1 ⊕ H 3 ⊕ G 1 ,

(1.2)

F1 ⊕ G 2 ⊕ H 2 , G4 ⊕ H1 , G 3 ⊕ H 2 , G 2 ⊕ H3 , G 1 ⊕ H4 and spherical functions are determined by F3 , Φ3 , Ψ3 , F2 ⊕ Φ1 , F2 ⊕ Ψ1 , Φ2 ⊕ F1 , Φ2 ⊕ Ψ1 , Ψ2 ⊕ F1 , Ψ2 ⊕ Φ1 , F1 ⊕ Φ1 ⊕ Ψ1 .

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47.5

2. The Selection of Spherical Functions for weight 5 1-For the quadratic form 2F3 = 2x21 + 2x1 x2 + 24x22 + 2x23 + 2x3 x4 + 24x24 + 2x25 + 2x5 x6 + 24x26    x1 2 1  x2  1 24      x3  2 1  , = (x1 , x2 , x3 , x4 , x5 , x6 )   x4   1 24     2 1  x5  x6 1 24 the determinant and some cofactors are D = 473 , A11 = 53016, A22 = 4418. By putting 2k = 6, Q = F3 , and appropriate i, j in theorem [1], we get 1 53016 8 2F3 = x21 − F3 ϕ11 = x21 − 6 473 47 which will be a spherical function of second order with respect to F3 . Similarly, 2G3 = 8x21 + 2x1 x2 + 10x22 + 8x23 + 2x3 x4 + 10x24 + 8x25 + 2x5 x6 + 10x26 , D = 473 , A11 = 62 410, A12 = −6241, 1 62 410 62 410 2G3 = x21 − G3 , ϕ11 = x21 − 6 473 311 469 ϕ12 = x1 x2 +

6241 1 6241 2G3 = x1 x2 + G3 , 3 6 47 311 469

Similarly, 2 (F2 ⊕ G1 ) = 2x21 + 2x1 x2 + 40x22 + 2x23 + 2x3 x4 + 40x24 + 8x25 + 2x5 x6 + 10x26 ,

D = 473 , A12 = −6241, A22 = 12482, A55 = 62 410, A56 = −6241, 1 6241 6241 ϕ12 = x1 x2 + 2 (F2 ⊕ Φ1 ) = x1 x2 + (F2 ⊕ G1 ) , 3 6 47 311 469 1 62410 62 410 ϕ55 = x27 − 2 (F2 ⊕ Φ1 ) = x25 − (F2 ⊕ G1 ) , 6 473 311 469 1 6241 6241 8 − ϕ56 = x5 x6 + 2 (F2 ⊕ Φ1 ) = x5 x6 + (F2 ⊕ G1 ) , 6 473 311 469 Similarly, 2 (F1 ⊕ G2 ) = 2x21 + 2x1 x2 + 40x22 + 8x23 + 2x3 x4 + 10x24 + 8x25 + 2x5 x6 + 10x26 , D = 473 , A11 = 249640, A12 = −6241, A56 = −6241, A66 = 49928. 1 249640 249 640 ϕ11 = x21 − 2 (F1 ⊕ G2 ) = x21 − (F1 ⊕ G2 ) , 6 473 311 469

˙ BARIS ¸ KENDIRL I˙

6

1 6241 6241 2 (F1 ⊕ G2 ) = x1 x2 + (F1 ⊕ G2 ) , 6 473 311 469 1 6241 6241 ϕ56 = x5 x6 + 2 (F1 ⊕ G2 ) = x5 x6 + (F1 ⊕ G2 ) , 6 473 311 469 1 49928 49 928 ϕ66 = x26 − 2 (F1 ⊕ G2 ) = x26 − (F1 ⊕ G2 ) , 3 6 47 311 469

ϕ12 = x1 x2 +

Similarly, 2H3 = 4x21 + 2x1 x2 + 12x22 + 4x23 + 2x3 x4 + 12x24 + 4x25 + 2x5 x6 + 12x26 , D = 473 , A11 = 26 508, A12 = −2209. 4 1 26 508 ϕ11 = x21 − 2H3 = x21 − H3 , 6 473 47 1 1 2209 ϕ12 = x1 x2 + 2H3 = x1 x2 + H3 , 6 473 141 Similarly, 2 (F2 ⊕ H1 ) = 2x21 + 2x1 x2 + 40x22 + 2x23 + 2x3 x4 + 40x24 + 4x25 + 2x5 x6 + 20x26 ,

D = 473 , A11 = 148 520, A12 = −3713, A55 = 74892, A56 = −6241, A66 = 24964. 1 3713 79 ϕ12 = x1 x2 + 2 (F2 ⊕ H1 ) = x1 x2 + (F2 ⊕ H1 ) , 3 6 47 6627 1 74892 24 964 ϕ55 = x25 − 2 (F2 ⊕ H1 ) = x25 − F2 ⊕ H 1 , 3 6 47 103 823 1 6241 6241 ϕ56 = x5 x6 + 2 (F2 ⊕ H1 ) = x5 x6 + F2 ⊕ H 1 , 3 6 47 311 469 Theorem 2.1. The set of following generalized 15 theta series  ∞ X  X 8 2 ΘF3 ,ϕ11 (q) = x1 − F3 q n , 47 n=1 F3 =n

∞ X  X

 62 410 ΘG3 ,ϕ11 (q) = − G3 q n , 311 469 n=1 G3 =n  ∞ X  X 6241 ΘG3 ,ϕ11 (q) = x1 x2 + G3 q n , 311 469 n=1 G3 =n  ∞ X X  6241 ΘF2 ⊕G1 ,ϕ11 (q) = x1 x2 + F2 ⊕ G1 q n , 311 469 n=1 F2 ⊕G1 =n   ∞ X X 62 410 2 ΘF2 ⊕G1 ,ϕ11 (q) = x5 − F2 ⊕ G 1 q n , 311 469 n=1 F2 ⊕G1 =n  ∞ X X  6241 ΘF2 ⊕G1 ,ϕ11 (q) = x5 x6 + F2 ⊕ G1 q n , 311 469 n=1 F2 ⊕G1 =n

x21

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47.7 ∞ X



 249 640 − ΘF1 ⊕G2 ,ϕ11 (q) = F1 ⊕ G 2 q n , 311 469 n=1 F1 ⊕G2 =n  ∞ X X  6241 ΘF1 ⊕G2 ,ϕ11 (q) = x1 x2 + F1 ⊕ G2 q n , 311 469 n=1 F1 ⊕G2 =n   ∞ X X 6241 ΘF1 ⊕G2 ,ϕ11 (q) = x5 x6 + F1 ⊕ G2 q n , 311 469 n=1 F1 ⊕G2 =n  ∞ X X  49 928 ΘF1 ⊕G2 ,ϕ11 (q) = x26 − F1 ⊕ G 2 q n , 311 469 n=1 F1 ⊕G2 =n   ∞ X X 4 2 ΘH3 ,ϕ11 (q) = x1 − H3 q n , 47 n=1 H3 =n  ∞ X  X 1 ΘH3 ,ϕ11 (q) = x1 x2 + H3 q n , 141 n=1 H3 =n   ∞ X X 79 ΘF2 ⊕H1 ,ϕ11 (q) = x1 x2 + F2 ⊕ H 1 q n , 6627 n=1 F2 ⊕H1 =n  ∞ X X  24 964 ΘF2 ⊕H1 ,ϕ11 (q) = F2 ⊕ H 1 q n , x25 − 103 823 n=1 F2 ⊕H1 =n   ∞ X X 6241 ΘF2 ⊕H1 ,ϕ11 (q) = x5 x6 + F2 ⊕ H 1 q n 311 469 n=1 F2 ⊕H1 =n
is a basis of S5 Γ0 (47) , χ−47 . X

x21

Proof. By similar calculations as in [1] , we get 1 ΘF3 ,ϕ11 (q) = (46q + 184q 2 + 184q 3 + 184q 4 + 920q 5 + 1104q 7 + 736q 8 + 47 2070q 9 + 1840q 10 + 2024q 11 − 322q 12 − 1472q 13 + 782q 14 − 3128q 15 + · · · ), 1 (−499280q 4 − 416454q 5 − 2788034q 8 − 3662936q 9 ΘG3 ,ϕ11 (q) = 311469 −2706370q 10 − 7157896q 12 − 10489528q 13 − 11488088q 14 − 7493848q 15 + · · · ), 1 ΘG3 ,ϕ12 (q) = (49928q 4 + 62410q 5 + 91922q 8 + 311469 449352q 9 + 582106q 10 − 31736q 12 + 467544q 13 + 2228568q 14 + 1829144q 15 + · · · ), 1 ΘF2 ⊕Φ1 ,ϕ12 (q) = (24964q+49928q 2 +37446q 3 +349496q 4 +748920q 5 +374460q 6 311469 +698992q 7 + 1897264q 8 + 1572732q 9 + 1497840q 10 +1647624q 11 + 1923390q 12 + 4595700q 13 + 3046770q 14 + 2001768q 15 + · · · ), 1 ΘF2 ⊕Φ1 ,ϕ55 (q) = (−249640q −499280q 2 +248478q 3 −1003208q 4 −4997448q 5 311 469 −3121662q 6 − 2006416q 7 − 10874446q 8 − 13235568q 9 − 9994896q 10 −9000984q 11 − 17988024q 12 − 38481744q 13 − 21746568q 14 − 27492936q 15 + · · · ), 1 ΘF2 ⊕Φ1 ,ϕ56 (q) = (24964q + 49928q 2 + 37446q 3 + 349496q 4 + 748920q 5 311469

8

˙ BARIS ¸ KENDIRL I˙

−248478q 6 − 1792760q 7 + 28450q 8 + 4064484q 9 + 1497840q 10 −3335880q 11 + 5038080q 12 + 10825080q 13 − 67920q 14 − 2981736q 15 + · · · ), 1 ΘF1 ⊕G2 ,ϕ11 (q) = (123658q − 2995680q 3 − 8996336q 4 − 7493848q 5 311469 −11982720q 6 − 40968848q 7 − 48957328q 8 − 47825814q 9 − 74938480q 10 −84924080q 11 − 131774574q 12 − 129411128q 13 − 88790014q 14 − 224815440q 15 + · · · ), 1 (12482q + 74892q 3 + 349496q 4 + 249640q 5 ΘF1 ⊕G2 ,ϕ12 (q) = 311469 +299568q 6 + 1397984q 7 + 1597696q 8 + 1460394q 9 + 2496400q 10 + 2746040q 11 +3870582q 12 + 4543448q 13 + 2173030q 14 + 4997448q 15 + · · · ), 1 ΘF1 ⊕G2 ,ϕ56 (q) = (12482q + 74892q 3 + 349496q 4 + 249640q 5 − 323370q 6 311469 +152108q 7 + 2220634q 8 + 1460394q 9 − 2487104q 10 + 1500164q 11 +8231148q 12 + 2051696q 13 − 941660q 14 + 6243324q 15 + · · · ), 1 ΘF1 ⊕G2 ,ϕ66 (q) = (−99856q − 599136q 3 − 2173030q 4 − 751244q 5 − 1773606q 6 311469 −8692120q 7 − 7175126q 8 − 6699648q 9 − 13741820q 10 − 13247188q 11 −25981152q 12 − 25134700q 13 − 14269550q 14 − 49946592q 15 + · · · ), 1 (46q 2 + 184q 4 + 40q 6 − 74q 7 − 208q 8 − 234q 9 + 712q 10 − 232q 11 ΘH3, ϕ11 (q) = 47 +616q 12 − 992q 13 − 568q 14 − 2000q 15 + · · · ), 1 ΘH3, ϕ11 (q) = (4q 2 + 16q 4 + 28q 6 − 80q 7 + 80q 8 141 −192q 9 + 160q 10 + 176q 11 − 20q 12 + 208q 13 − 736q 14 − 272q 15 + · · · ), 1 ΘF2 ⊕H1 ,ϕ11 (q) = (316q+948q 2 +1896q 3 +3792q 4 +3160q 5 +4740q 6 +14378q 7 +13904q 8 6627 +15642q 9 +31600q 10 +34760q 11 +34146q 12 +31004q 13 +22174q 14 +15276q 15 +· · · ), 1 ΘF2 ⊕H1 ,ϕ55 (q) = (−10267493488q−9451695806q 2 +23798177704q 3 −37806783224q 4 103 823 −102674934880q 5 − 68609263688q 6 − 275013891782q 7 − 280963436208q 8 −59874449838q 9 − 514330517008q 10 − 873214867784q 11 − 771495775512q 12 −758443012600q 13 − 1329203614592q 14 − 2482426075976q 15 + · · · ), 1 ΘF2 ⊕H1 ,ϕ56 (q) = (24964q+74892q 2 +149784q 3 +299568q 4 +249640q 5 +374460q 6 311 469 +512924q 7 − 1393336q 8 − 633096q 9 + 4988152q 10 + 2746040q 11 −2484780q 12 + 2051696q 13 + 4892944q 14 + 49974484q 15 + · · · ). Since the 15th determinant of the coefficients of the theta series is different from
zero, the 15 generalized theta series in the Theorem is a basis of S5 Γ0 (47) , χ−47 .

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47.9

Theorem 2.2. The theta series ΘQ (q) = 1 +

∞ X

r (n; Q) q n , q = e2πiτ

n=1

is given as linear combination of two Eisenstein series and cusp forms as 2 G5,47 + aH5,47 + 169920 +c1 ΘF3 ,ϕ11 +c2 ΘG3 ,ϕ11 (q)+c3 ΘG3 ,ϕ12 (q)+c4 ΘF2 ⊕Φ1 ,ϕ12 (q)+c5 ΘF2 ⊕Φ1 ,ϕ55 (q)+c6 ΘF2 ⊕Φ1 ,ϕ56 (q) +c7 ΘF1 ⊕G2 ,ϕ11 (q) + c8 ΘF1 ⊕G2 ,ϕ12 (q) + c9 ΘF1 ⊕G2 ,ϕ56 (q) + c10 ΘF1 ⊕G2 ,ϕ66 (q) +c11 ΘH3, ϕ11 (q)+c12 ΘH3, ϕ11 (q)+c13 ΘF2 ⊕H1 ,ϕ11 (q)+c14 ΘF2 ⊕H1 ,ϕ55 (q)+c15 ΘF2 ⊕H1 ,ϕ56 (q) . Consequently, the representation number of integer n by Q is given by taking the sum of the coefficients of q n .The constants a, c1 , c2 , · · · , c15 are given in the table [4]. Proof. By Corollary 2.7 in [5] , the Eisenstein subspace is generated by two linearly independent Eisenstein series  
∞ X X L 1 − 5, χ−47  + χ−47 (d) d5−1  q n G5,47 (q) = 2 n=1 d>0,d|n

=

169920 + q + 17q 2 + 82q 3 + 273q 4 − 624q 5 + 1394q 6 + 2402q 7 + 4369q 8 2


+6643q 9 −10608q 10 −14640q 11 +1650q 12 −28560q 13 +40834q 14 −51168q 15 +· · · O q 16 ,   ∞ X X  H5,47 (q) = χ−47 (n/d) d5−1  q n n=1

d>0,d|n

= q + 17q 2 + 82q 3 + 273q 4 + 624q 5 + 1394q 6 + 2402q 7 + 4369q 8 + 6643q 9
+10608q 10 + 14640q 11 + 22386q 12 + 28560q 13 + 40834q 14 + 51168q 15 + · · · O q 16 . Consequently, the theta series ΘQ (q) = 1 +

∞ X

r (n; Q) q n , q = e2πiτ

n=1

is given as linear combination of two Eisenstein series and cusp forms as 2 G5,47 + aH5,47 + 169920 +c1 ΘF3 ,ϕ11 +c2 ΘG3 ,ϕ11 (q)+c3 ΘG3 ,ϕ12 (q)+c4 ΘF2 ⊕Φ1 ,ϕ12 (q)+c5 ΘF2 ⊕Φ1 ,ϕ55 (q)+c6 ΘF2 ⊕Φ1 ,ϕ56 (q) +c7 ΘF1 ⊕G2 ,ϕ11 (q) + c8 ΘF1 ⊕G2 ,ϕ12 (q) + c9 ΘF1 ⊕G2 ,ϕ56 (q) + c10 ΘF1 ⊕G2 ,ϕ66 (q) +c11 ΘH3, ϕ11 (q)+c12 ΘH3, ϕ11 (q)+c13 ΘF2 ⊕H1 ,ϕ11 (q)+c14 ΘF2 ⊕H1 ,ϕ55 (q)+c15 ΘF2 ⊕H1 ,ϕ56 (q) .

˙ BARIS ¸ KENDIRL I˙

10

2.1. Weight 6 Case. By Theorem [see 1 ] the theta series of quadratic forms in [2.1] are in M6 (Γ0 (47)) . In this paper, we will obtain the formulas of representation number for the following 27 different quadratic forms of 12 variables Q = F6 , G6 , H6 , F5 ⊕ G1 , F5 ⊕ H1 , F4 ⊕ G2 , F4 ⊕ H2 , F4 ⊕ G1 ⊕ H1 F3 ⊕ G 3 , F 3 ⊕ H 3 , F 3 ⊕ G 2 ⊕ H 1 , F 3 ⊕ H 2 ⊕ G 1 , F 3 ⊕ H 2 ⊕ G 1 , F2 ⊕ G 4 , F 2 ⊕ H 4 , F 2 ⊕ G 3 ⊕ H 1 , F 2 ⊕ H 3 ⊕ G 1 , F 1 ⊕ G 5 ,

(2.1)

F1 ⊕ H 5 , F 1 ⊕ G 4 ⊕ H 1 , F 1 ⊕ H 4 ⊕ G 1 , F 1 ⊕ G 3 ⊕ H 2 , F1 ⊕ H3 ⊕ G2 , G5 ⊕ H1 , G4 ⊕ H2 , G3 ⊕ H3 , G2 ⊕ H4 , G1 ⊕ H5 . 0

In these formulas one can replace G1 , and H1 by G1 , and H10 respectively. We have the following theorem for the Eisenstein part of theta series associated to the quadratic form. Theorem 2.3. Let Q be a positive definite form of 12 variables whose theta series ΘQ are in M6 (Γ0 (47)) . Then the Eisenstein part of ΘQ is E (q; Q) = 1 +

∞ X

ασ 5 (n) q n + βσ 5 (n) q 47n




n=1

respectively, where ρ6 = −

5!

6ζ

(2π)

(6) = −

1 504

473 + 1 252 = , 476 − 1 51911 476 + 473 26 163 396 β = −504 =− 6 47 − 1 51 911 α = 504

and EF6 ( q) = · · · = EH6 = 1+

∞ ∞
252 X n 252 X ∗ q − 473 q 47n σ 5 (n) = 1+ σ (n) q n 51911 n=1 51911 n=1 5

252 8316 2 1008 3 266364 4 787752 5 33264 6 q + q + q + q + q + q 51911 51911 851 51911 51911 851 4235616 7 8523900 8 14941836 9 25995816 10 40585104 11 q + q + q (2.2) + q + q + 51911 51911 51911 51911 51911 1065456 12 93566088 13 139775328 14 3151008 15 272765052 16 + q + q + q + q + q 851 51911 51911 851 51911 357804216 17 493080588 18 623977200 19 + q + q + q + ··· 51911 51911 51911  σ 5 (n) if n ≥ 1 and 47 - n σ ∗5 (n) = . σ 5 (n) − 473 σ 5 (n/47) if 47/ | n =1+

Proof. See [1] .

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47. 11

3. The Selection of Spherical Functions for weight 6 1-For quadratic form 2F4 = 2x21 +2x1 x2 +24x22 +2x23 +2x3 x4 +24x24 +2x25 +2x5 x6 +24x26 +2x27 +2x7 x8 +24x28    2 1 x1 1 24  x2       x3  2 1      x4  1 24  , = (x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 )    x5  2 1      x6  1 24     2 1  x7  1 24 x8 the determinant and some cofactors are D = 474 , A12 = −103 823, A22 = 207 646. By putting 2k = 8, Q = F4 , and appropriate i, j in theorem [1], we get 1 1 103 823 2F4 = x1 x2 + F4 ϕ12 = x1 x2 + 8 474 188 1 207 646 1 ϕ22 = x22 − 2F4 = x22 − F4 , 4 8 47 94 which will be spherical functions of second order with respect to F4 .Similarly, 2G4 = 6x21 +2x1 x2 +8x22 +6x23 +2x3 x4 +8x24 +6x25 +2x5 x6 +8x26 +6x27 +2x7 x8 +8x28 D = 474 , A11 = 830584, A12 = −103 823, 1 830 584 2 2Φ2 = x21 − G2 , ϕ11 = x21 − 8 474 47 1 103 823 1 ϕ12 = x1 x2 + 2Φ2 = x1 x2 + G2 , 4 8 47 188 2 (F3 ⊕ G1 ) = 2x21 +2x1 x2 +24x22 +2x23 +2x3 x4 +24x24 +2x25 +2x5 x6 +24x26 +8x27 +2x7 x8 +10x28 D = 474 , A12 = −103823, A22 = 207 646, A77 = 830 584, A78 = −103823, 1 103 823 1 ϕ12 = x1 x2 + 2 (F3 ⊕ Φ1 ) = x1 x2 + (F3 ⊕ G1 ) , 4 8 47 188 1 830 584 2 ϕ77 = x27 − 2 (F3 ⊕ Φ1 ) = x27 − (F3 ⊕ G1 ) , 4 8 47 47 1 103 823 1 ϕ78 = x7 x8 + 2 (F3 ⊕ Φ1 ) = x7 x8 + (F3 ⊕ G1 ) , 8 474 188 1 622 938 3 ϕ88 = x28 − 2 (F3 ⊕ Φ1 ) = x28 − (F3 ⊕ G1 ) , 8 474 94 2 2 2 2 2 2 (F2 ⊕ G2 ) = 2x1 +2x1 x2 +24x2 +2x3 +2x3 x4 +24x4 +6x5 +2x5 x6 +8x26 +6x27 +2x7 x8 +8x28 , D = 474 , A11 = 2491 752, A12 = −103 823, A56 = −103 823, A66 = 622 938, 1 2491 752 6 ϕ11 = x21 − 2 (F2 ⊕ G2 ) = x21 − (F2 ⊕ G2 ) , 8 474 47 1 103 823 1 ϕ12 = x1 x2 + 2 (F2 ⊕ G2 ) = x1 x2 + (F2 ⊕ G2 ) , 8 474 188 1 103 823 1 ϕ56 = x5 x6 + 2 (F2 ⊕ G2 ) = x5 x6 + (F2 ⊕ G2 ) , 8 474 188

˙ BARIS ¸ KENDIRL I˙

12

1 622 938 3 2 (F2 ⊕ G2 ) = x26 − (F2 ⊕ G2 ) , 4 8 47 94 2 2 2 2 2 2 (F1 ⊕ G3 ) = 2x1 +2x1 x2 +24x2 +6x3 +2x3 x4 +8x4 +6x5 +2x5 x6 +8x26 +6x27 +2x7 x8 +8x28 , D = 474 , A11 = 2491 752, A12 = −103 823, A33 = 830 584, A44 = 622 938, 1 2491 752 6 ϕ11 = x21 − 2 (F1 ⊕ G3 ) = x21 − (F1 ⊕ G3 ) , 8 474 47 1 1 103 823 ϕ12 = x1 x2 + 2 (F1 ⊕ G3 ) = x1 x2 + (F1 ⊕ G3 ) , 8 474 188 1 830 584 2 2 (F1 ⊕ G3 ) = x23 − (F1 ⊕ G3 ) , ϕ33 = x23 − 8 474 47 1 622 938 3 ϕ44 = x24 − 2 (F1 ⊕ G3 ) = x24 − (F1 ⊕ G3 ) , 4 8 47 94 2H4 = 4x21 +2x1 x2 +12x22 +4x23 +2x3 x4 +12x24 +4x25 +2x5 x6 +12x26 +4x27 +2x7 x8 +12x28 , D = 474 , A11 = 1245 876, A12 = −103 823, 1 1245876 3 ϕ11 = x21 − 2 (F3 ⊕ H1 ) = x21 − (F3 ⊕ H1 ) , 4 8 47 47 1 103 823 1 ϕ12 = x1 x2 + 2 (F3 ⊕ H1 ) = x1 x2 + (F3 ⊕ H1 ) , 8 474 188 ϕ66 = x26 −

2 (F3 ⊕ H1 ) = 2x21 +2x1 x2 +24x22 +2x23 +2x3 x4 +24x24 +2x25 +2x5 x6 +24x26 +4x27 +2x7 x8 +12x28 , ϕ11

D = 474 , A11 = 2491 752, 6 1 2491 752 2 (F3 ⊕ H1 ) = x21 − (F3 ⊕ H1 ) , = x21 − 4 8 47 47

4. The Solutions of Q = n and the Theta Series associated to the Quadratic Forms 1-F1 = x21 + x1 x2 + 12x22 = n has the following solutions: n = 1 =⇒ x1 = ±1, x2 = 0, n = 4 =⇒ x1 = ±2, x2 = 0, n = 9 =⇒ x1 = ±3, x2 = 0, n = 12 =⇒ x1 = 1, x2 = −1, or x1 = −1, x2 = 1, or x1 = 0, x2 = −1, or x1 = 0, x2 = 1, n = 14 =⇒ x1 = −2, x2 = 1, or x1 = 2, x2 = −1, or x1 = −1, x2 = −1, or x1 = 1, x2 = 1, n = 16 =⇒ x1 = ±4, x2 = 0, n = 18 =⇒ x1 = −3, x2 = 1, or x1 = 3, x2 = −1, or x1 = −2, x2 = −1 or x1 = 2, x2 = 1, or x1 = 2, x2 = −1 or x1 = −2, x2 = 1, There is no integral solutions for n=2, 3, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19. 2-G1 = 3x21 + x1 x2 + 4x22 = n has the following solutions: n = 3 =⇒ x1 = ±1, x2 = 0, n = 4 =⇒ x1 = 0, x2 = ±1, n = 6 =⇒ x1 = 1, x2 = −1, or x1 = −1, x2 = 1 n = 8 =⇒ x1 = −1, x2 = −1 or x1 = 1, x2 = 1,

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47. 13

n = 12 =⇒ x1 = −2, x2 = 0 or x1 = 2, x2 = 0, n = 14 =⇒ x1 = −2, x2 = 1 or x1 = 2, x2 = −1, n = 16 =⇒ x1 = 0, x2 = ±2, n = 17 =⇒ x1 = −1, x2 = 2 or x1 = 1, x2 = −2, n = 18 =⇒ x1 = −2, x2 = −1 or x1 = 2, x2 = 1, There is no integral solutions for n=1, 2, 5, 7, 9, 10, 11, 13, 15, 19. 3-H1 = 2x21 + x1 x2 + 6x22 = n has the following solutions: n = 2 =⇒ x1 = ±1, x2 = 0, n = 6 =⇒ x1 = 0, x2 = ±1, n = 7 =⇒ x1 = −1, x2 = 1, or x1 = 1, x2 = −1 n = 8 =⇒ x1 = ±2, x2 = 0, n = 9 =⇒ x1 = −1, x2 = −1, or x1 = 1, x2 = 1, n = 12 =⇒ x1 = −2, x2 = 1, or x1 = 2, x2 = −1, n = 16 =⇒ x1 = −2, x2 = −1, x1 = 2, x2 = 1, n = 18 =⇒ x1 = ±3, x2 = 0, There is no integral solutions for n=1, 3, 4, 5, 10, 11, 13, 14, 15, 17, 19. ΘF1 (q) = 1 + 2q + 2q 4 + 2q 9 + 4q 12 + 4q 14 + 2q 16 + 4q 18 + · · · ,

ΘG1 (q) = 1 + 2q 3 + 2q 4 + 2q 6 + 2q 8 + 2q 12 + 2q 14 + 2q 16 + 2q 17 + 2q 18 + · · · ΘH1 (q) = 1+4q 2 +4q 4 +4q 6 +4q 7 +12q 8 +12q 9 +8q 10 +8q 11 +8q 12 +8q 13 +20q 14 +16q 15 +16q 16 + 8q 17 + 24q 18 + 8q 19 + · · · , ΘF6 (q) = 1+12q+60q 2 +160q 3 +252q 4 +312q 5 +544q 6 +960q 7 +1020q 8 +876q 9 +1560q 10 +2400q 11 +2104q 12 +2280q 13 +4248q 14 +6320q 15 +7212q 16 +8088q 17 +12324q 18 17808q 19 +· · · ΘG6 (q) = 1+12q 3 +12q 4 +72q 6 +120q 7 +72q 8 +280q 9 +600q 10 +600q 11 +1072q 12 + +1920q 13 + 2532q 14 + 3672q 15 + 5232q 16 + 6852q 17 + 10076q 18 + 12504q 19 + · · · ΘH6 (q) = 1+12q 2 +60q 4 +172q 6 +12q 7 +372q 8 +132q 9 +792q 10 +600q 11 +1576q 12 +1560q 13 + 2700q 14 + 3120q 15 + 4896q 16 + 6264q 17 + 9496q 18 + 11544q 19 + · · · ΘF5 ⊕G1 (q) = 1+10q+40q 2 +82q 3 +112q 4 +212q 5 +482q 6 +680q 7 +686q 8 +1134q 9 +1940q 10 +1984q 11 +1982q 12 +3224q 13 +4502q 14 +5040q 15 +5752q 16 +8146q 17 +12502q 18 +14504q 19 +· · · ΘF4 ⊕G2 (q) = 1+8q+24q 2 +36q 3 +60q 4 +176q 5 +328q 6 +360q 7 +576q 8 +1200q 9 +1496q 10 +1544q 11 +2496q 12 +3632q 13 +4060q 14 +4744q 15 +6184q 16 +8796q 17 +11004q 18 +11816q 19 +· · · ΘF5 ⊕H1 (q) = 1+10q+42q 2 +100q 3 +170q 4 +272q 5 +422q 6 +566q 7 +782q 8 +1152q 9 +1400q 10 + +1704q 11 +2586q 12 +3384q 13 +3964q 14 +5380q 15 +7132q 16 +8984q 17 +11182q 18 +13420q 19 +· · · ΘF4 ⊕H2 (q) = 1+8q+28q 2 +64q 3 +124q 4 +208q 5 +292q 6 +420q 7 +644q 8 +884q 9 +1240q 10

14

˙ BARIS ¸ KENDIRL I˙

+1800q 11 +2296q 12 +3032q 13 +4228q 14 +5264q 15 +6632q 16 +8664q 17 +10784q 18 +13640q 19 +· · · ΘF4 ⊕G1 ⊕H1 (q) = 1+8q+26q 2 +50q 3 +90q 4 +180q 5 +296q 6 +434q 7 +704q 8 +990q 9 +1212q 10 +1800q 11 +2528q 12 +2936q 13 +3902q 14 +5388q 15 +6696q 16 +8550q 17 +10648q 18 +13516q 19 +· · · ΘF3 ⊕G3 (q) = 1+6q+12q 2 +14q 3 +48q 4 +132q 5 +162q 6 +216q 7 +570q 8 +890q 9 +924q 10 +1536q 11 +2582q 12 +3048q 13 +3738q 14 +5080q 15 +6756q 16 +8766q 17 +9782q 18 +12216q 19 +· · · ΘF3 ⊕H3 (q) = 1+6q+18q 2 +44q 3 +90q 4 +144q 5 +218q 6 +330q 7 +462q 8 +712q 9 +1104q 10 +1512q 11 +2114q 12 +2880q 13 +3672q 14 +4932q 15 +6528q 16 +8256q 17 +10578q 18 +13452q 19 +· · · ΘF3 ⊕G2 ⊕H1 (q) = 1+6q+14q 2 +24q 3 +58q 4 +120q 5 +182q 6 +318q 7 +546q 8 +724q 9 +1080q 10 +1712q 11 +2122q 12 +2752q 13 +4016q 14 +5068q 15 +6384q 16 +8332q 17 +10414q 18 +13604+· · · ΘF3 ⊕G1 ⊕H2 (q) = 1+6q+16q 2 +34q 3 +72q 4 +124q 5 +198q 6 +348q 7 +526q 8 +726q 9 +1108q 10 +1528q 11 +2006q 12 +2920q 13 +3866q 14 +4864q 15 +6572q 16 +8434q 17 +10578q 18 +13472q 19 +· · · ΘF2 ⊕G4 (q) = 1+4q +4q 2 +8q 3 +44q 4 +72q 5 +64q 6 +208q 7 +452q 8 +468q 9 +728q 10 +1520q 11 +1960q 12 +2424q 13 +3680q 14 +4992q 15 +6612q 16 +8192q 17 +9764q 18 +13696q 19 +· · · ΘF2 ⊕H4 (q) = 1+4q+12q 2 +32q 3 +60q 4 +104q 5 +168q 6 +232q 7 +364q 8 +572q 9 +856q 10 +1328q 11 +1864q 12 +2472q 13 +3328q 14 +4432q 15 +5924q 16 +7928q 17 +10396q 18 +13152q 19 +· · · ΘF2 ⊕G3 ⊕H1 (q) = 1+4q+6q 2 +14q 3 +42q 4 +68q 5 +112q 6 +242q 7 +364q 8 +522q 9 +948q 10 +1336q 11 +1712q 12 +2680q 13 +3666q 14 +4580q 15 +6268q 16 +8082q 17 +10336q 18 +13388q 19 +· · · ΘF2 ⊕G1 ⊕H3 (q) = 1+4q+10q 2 +26q 3 +50q 4 +84q 5 +156q 6 +246q 7 +376q 8 +614q 9 +868q 10 +1224q 11 +1824q 12 +2480q 13 +3394q 14 +4700q 15 +6084q 16 +7870q 17 +10284q 18 +13148q 19 +· · · ΘF1 ⊕G5 (q) = 1 + 2q + 10q 3 + 32q 4 + 20q 5 + 50q 6 + 230q 8 + 262q 9 + 740q 10 +1120q 11 +1294q 12 +2200q 13 +3294q 14 +4352q 15 +6016q 16 +7530q 17 +10214q 18 +13864q 19 +· · · ΘF1 ⊕H5 (q) = 1+2q+10q 2 +20q 3 +42q 4 +80q 5 +110q 6 +190q 7 +270q 8 +432q 9 +712q 10 +1064q 11 +1594q 12 +2120q 13 +2948q 14 +3780q 15 +5412q 16 +7240q 17 +9974q 18 +12524q 19 +· · · ΘF1 ⊕G4 ⊕H1 (q) = 1+2q+2q 2 +12q 3 +26q 4 +32q 5 +86q 6 +166q 7 +214q 8 +424q 9 +728q 10 +920q 11 +1498q 12 +2344q 13 +2980q 14 +4212q 15 +5908q 16 +7552q 17 +10110q 18 +12860q 19 +· · · ΘF1 ⊕G1 ⊕H4 (q) = 1+2q+8q 2 +18q 3 +32q 4 +68q 5 +106q 6 +176q 7 +302q 8 +446q 9 +692q 10

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47. 15

+1040q 11 +1470q 12 +2104q 13 +3022q 14 +4080q 15 +5568q 16 +7314q 17 +9710q 18 +12248q 19 +· · · ΘF1 ⊕G3 ⊕H2 (q) = 1+2q+4q 2 +14q 3 +24q 4 +44q 5 +102q 6 +156q 7 +250q 8 +466q 9 +668q 10 +984q 11 +1582q 12 +2136q 13 +2958q 14 +4296q 15 +5656q 16 +7406q 17 +9970q 18 +12720q 19 +· · · ΘF1 ⊕G2 ⊕H3 (q) = 1+2q+6q 2 +16q 3 +26q 4 +56q 5 +106q 6 +162q 7 +290q 8 +460q 9 +656q 10 +1040q 11 +1522q 12 +2088q 13 +3036q 14 +4140q 15 +5524q 16 +7428q 17 +9906q 18 +12548q 19 +· · · ΘG5 ⊕H1 (q) = 1+2q 2 +10q 3 +10q 4 +20q 5 +72q 6 +82q 7 +152q 8 +342q 9 +460q 10 +680q 11 +1312q 12 +1720q 13 +2470q 14 +3772q 15 +5152q 16 +6814q 17 +9592q 18 +11980q 19 +· · · ΘG4 ⊕H2 (q) = 1 + 4q 2 + 8q 3 + 12q 4 + 32q 5 + 68q 6 + 84q 7 + 204q 8 + 316q 9 + 472q 10 +792q 11 +1256q 12 +1640q 13 +2688q 14 +3616q 15 +5008q 16 +6752q 17 +9440q 18 +11768q 19 +· · · ΘG3 ⊕H3 (q) = 1 + 6q 2 + 6q 3 + 18q 4 + 36q 5 + 68q 6 + 102q 7 + 228q 8 + 290q 9 + 540q 10 +792q 11 +1232q 12 +1680q 13 +2694q 14 +3412q 15 +5112q 16 +6618q 17 +9332q 18 +11484q 19 +· · · ΘG2 ⊕H4 (q) = 1 + 8q 2 + 4q 3 + 28q 4 + 32q 5 + 80q 6 + 112q 7 + 240q 8 + 288q 9 + 600q 10 +728q 11 +1296q 12 +1648q 13 +2724q 14 +3304q 15 +5216q 16 +6364q 17 +9332q 18 +11320q 19 +· · · ΘG1 ⊕H5 (q) = 1 + 10q 2 + 2q 3 + 42q 4 + 20q 5 + 112q 6 + 90q 7 + 272q 8 + 270q 9 + 652q 10 +680q 11 +1392q 12 +1544q 13 +2806q 14 +3180q 15 +5376q 16 +6102q 17 +9376q 18 +11020q 19 +· · · Now we will determine a basis of S6 (Γ0 (47)) . Theorem 4.1. The set of following generalized 19 theta series is a basis of S6 (Γ0 (47))  ∞ X  X 1 F4 q n , ΘF4 ,ϕ12 (q) = x1 x2 + 188 n=1 F4 =n

∞ X  X

 1 ΘF4 ,ϕ22 (q) = − F4 q n , 94 n=1 F4 =n  ∞ X  X 2 ΘG4 ,ϕ11 (q) = x21 − G2 q n , 47 n=1 G4 =n   ∞ X X 1 ΘG4 ,ϕ12 (q) = x1 x2 + G2 q n , 188 n=1 G2 =n  ∞ X X  1 ΘF3 ⊕G1 ,ϕ12 (q) = x1 x2 + F3 ⊕ G1 q n , 188 n=1 F3 ⊕G1 =n

x22

˙ BARIS ¸ KENDIRL I˙

16

∞ X



 2 − F3 ⊕ G 1 q n , ΘF3 ⊕G1 ,ϕ77 (q) = 47 n=1 F3 ⊕G1 =n  ∞ X X  1 ΘF3 ⊕G1 ,ϕ78 (q) = x7 x8 + F3 ⊕ G1 q n , 188 n=1 F3 ⊕G1 =n   ∞ X X 3 ΘF3 ⊕G1 ,ϕ88 (q) = x28 − F3 ⊕ G1 q n , 94 n=1 F3 ⊕G1 =n   ∞ X X 6 2 ΘF2 ⊕G2 ,ϕ11 (q) = x1 − F2 ⊕ G2 q n , 47 n=1 F2 ⊕G2 =n   ∞ X X 1 ΘF2 ⊕G2 ,ϕ12 (q) = x1 x2 + F2 ⊕ G2 q n , 188 n=1 F2 ⊕G2 =n   ∞ X X 1 ΘF2 ⊕G2 ,ϕ56 (q) = x5 x6 + F2 ⊕ G2 q n , 188 n=1 F2 ⊕G2 =n  ∞ X X  3 ΘF2 ⊕G2 ,ϕ66 (q) = x26 − F2 ⊕ G2 q n , 94 n=1 F2 ⊕G2 =n   ∞ X X 6 2 ΘF1 ⊕G3 ,ϕ11 (q) = x1 − F1 ⊕ G3 q n , 47 n=1 F1 ⊕G3 =n  ∞ X X  1 F1 ⊕ G3 q n , ΘF1 ⊕G3 ,ϕ12 (q) = x1 x2 + 188 n=1 F1 ⊕G3 =n   ∞ X X 2 ΘF1 ⊕G3 ,ϕ33 (q) = x23 − F1 ⊕ G3 q n , 47 n=1 F1 ⊕G3 =n   ∞ X X 3 2 (F1 ⊕ G3 ) q n , ΘF1 ⊕G3 ,ϕ44 (q) = x4 − 94 n=1 F1 ⊕G3 =n  ∞ X  X 3 ΘH4 ,ϕ11 (q) = x21 − H4 q n , 47 n=1 H4 =n   ∞ X X 1 ΘH4 ,ϕ12 (q) = x1 x2 + H4 q n , 188 n=1 H4 =n  ∞ X X  6 ΘF3 ⊕H1 ,ϕ11 (q) = x21 − F3 ⊕ H1 q n . 47 n=1 X

x27

F3 ⊕H1 =n

Proof. By similar calculations as in [1] , we get  ∞ X  X 1 1 ΘF4 ,ϕ12 (q) = x1 x2 + F4 q n = (2q + 12q 2 + 24q 3 + 24q 4 188 47 n=1 F4 =n

5

6

+60q + 144q + 112q 7 + 48q 8 + 234q 9 + 360q 10 + 264q 11 + 242q 12 + 112q 13 +178q 14 + 244q 15 − 444q 16 − 220q 17 + 722q 18 + 220q 19 + · · · ),  ∞ X  X 1 1 ΘF4 ,ϕ22 (q) = x22 − F4 q n = (90q + 540q 2 + 1080q 3 + 1080q 4 94 47 n=1 F4 =n

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47. 17

+2700q 5 + 6480q 6 + 5040q 7 + 2160q 8 + 10530q 9 + 16200q 10 + 11880q 11 + 12958q 12 +17448q 13 +34894q 14 +39932q 15 +17244q 16 +56276q 17 +96598q 18 +71940q 19 +· · · ),  ∞ X  X 2 1 x21 − G2 q n = ΘG4 ,ϕ11 (q) = (46q 3 − 64q 4 + 274q 6 − 108q 7 47 47 n=1 G4 =n

−418q 8 + 816q 9 − 60q 10 − 912q 11 + 264q 12 + 112q 13 − 1360q 14 + 556q 15 − 2388q 16 66q 17 + 1956q 18 − 852q 19 + · · · ),   ∞ X X 1 1 x1 x2 + ΘG4 ,ϕ12 (q) = G2 q n = (6q 3 +8q 4 −46q 6 +84q 7 +158q 8 −384q 9 188 47 n=1 G2 =n

−204q 10 + 960q 11 − 456q 12 − 1424q 13 + 1016q 14 + 988q 15 − 900q 16 − 678q 17 108q 18 + 3420q 19 + · · · ),  ∞ X X  1 1 x1 x2 + ΘF3 ⊕G1 ,ϕ12 (q) = F3 ⊕ G 1 q n = (−3q − 12q 2 − 15q 3 188 94 n=1 F3 ⊕G1 =n

−40q 4 − 150q 5 − 198q 6 − 140q 7 − 392q 8 − 693q 9 − 540q 10 − 616q 11 − 1184q 12 −2312q 13 − 3054q 14 − 2808q 15 − 6552q 16 − 10753q 17 − 10046q 18 − 15492q 19 + · · · ),  ∞ X X  2 1 2 ΘF3 ⊕G1 ,ϕ77 (q) = x7 − (F3 ⊕ G1 ) q n = (−12q − 48q 2 47 47 n=1 F3 ⊕G1 =n

34q 3 + 404q 4 + 528q 5 + 54q 6 + 568q 7 + 1910q 8 + 800q 9 − 468q 10 + 1672q 11 +2032q 12 +528q 13 +2072q 14 +3244q 15 +4248q 16 +3894q 17 −516q 18 +2140q 19 +· · · ),  ∞ X X  1 1 F3 ⊕ G 1 q n = (3q + 12q 2 ΘF3 ⊕G1 ,ϕ78 (q) = x7 x8 + 188 94 n=1 F3 ⊕G1 =n

+15q 3 + 40q 4 + 150q 5 + 10q 6 − 988q 7 − 1676q 8 + 317q 9 + 1668q 10 − 2392q 11 − 2388q 12 +6072q 13 + 2994q 14 − 6216q 15 − 3600q 16 + 3609q 17 + 1962q 18 − 13084q 19 + · · · ),  ∞ X X  1 3 2 (F3 ⊕ G1 ) q n = (−9q − 36q 2 ΘF3 ⊕G1 ,ϕ88 (q) = x8 − 94 47 n=1 F3 ⊕G1 =n

−45q 3 − 26q 4 + 114q 5 + 628q 6 + 896q 7 + 610q 8 + 1493q 9 + 2328q 10 + 1160q 11 +960q 12 + 396q 13 − 608q 14 + 600q 15 − 2736q 16 − 2367q 17 + 5676q 18 + 2216q 19 + · · · ),  ∞ X X  6 1 ΘF2 ⊕G2 ,ϕ11 (q) = (70q+140q 2 −72q 3 +176q 4 x21 − F2 ⊕ G2 q n = 47 47 n=1 F2 ⊕G2 =n

+868q 5 − 112q 6 − 96q 7 + 2320q 8 + 1030q 9 + 424q 10 + 3648q 11 + 3718q 12 + 6856q 13 +7046q 14 + 2372q 15 + 11972q 16 + 11980q 17 − 3842q 18 − 180q 19 + · · · ),  ∞ X X  1 1 ΘF2 ⊕G2 ,ϕ12 (q) = x1 x2 + F2 ⊕ G2 q n = (q + 2q 2 + 3q 3 + 24q 4 188 47 n=1 5

6

F2 ⊕G2 =n 7

+50q + 36q + 98q + 248q 8 + 243q 9 + 280q 10 + 506q 11 + 554q 12 + 670q 13 +760q 14 + 426q 15 + 684q 16 + 997q 17 + 70q 18 − 110q 19 + · · · ),  ∞ X X  1 1 ΘF2 ⊕G2 ,ϕ56 (q) = x5 x6 + F2 ⊕ G2 q n = (q + 2q 2 + 3q 3 + 24q 4 118 47 n=1 F2 ⊕G2 =n

+50q 5 − 58q 6 − 278q 7 − 34q 8 + 431q 9 − 660q 10 − 1562q 11 + 1024q 12 +1610q 13 − 1966q 14 − 890q 15 + 2564q 16 + 1937q 17 − 1904q 18 − 5186q 19 + · · · ),

˙ BARIS ¸ KENDIRL I˙

18

ΘF2 ⊕G2 ,ϕ66 (q) =

∞ X

X



n=1 F2 ⊕G2 =n

x26

 3 1 − F2 ⊕ G 2 q n = (−6q−12q 2 −18q 3 −50q 4 94 47

+76q 5 + 254q 6 − 24q 7 + 298q 8 + 1362q 9 + 952q 10 + 912q 11 + 1940q 12 + 1244q 13 +1738q 14 + 828q 15 − 2224q 16 + 974q 17 + 426q 18 − 4604q 19 + · · · ),  ∞ X X  1 6 (70q+140q 2 −72q 3 +176q 4 ΘF1 ⊕G3 ,ϕ11 (q) = x21 − F1 ⊕ G3 q n = 47 47 n=1 F1 ⊕G3 =n

5

6

+868q − 112q − 96q 7 + 2320q 8 + 1030q 9 + 424q 10 + 3648q 11 + 3718q 12 + 6856q 13 +7046q 14 + 2372q 15 + 12164q 16 + 12200q 17 − 3842q 18 + 732q 19 + · · · ),  ∞ X X  1 1 (F1 ⊕ G3 ) q n = (q + 2q 2 + 3q 3 + ΘF1 ⊕G3 ,ϕ12 (q) = x1 x2 + 188 47 n=1 F1 ⊕G3 =n

+24q 4 + 50q 5 + 36q 6 + 98q 7 + 248q 8 + 243q 9 + 280q 10 + 506q 11 + 554q 12 +670q 13 + 760q 14 + 426q 15 + 676q 16 + 980q 17 + 70q 18 − 148q 19 + · · · ),  ∞ X X  2 1 ΘF1 ⊕G3 ,ϕ33 (q) = x23 − F1 ⊕ G3 q n = (86q+172q 2 −24q 3 +560q 4 47 47 n=1 F1 ⊕G3 =n

5

6

+1668q +464q +1472q 7 +6288q 8 +4918q 9 +4904q 10 +11744q 11 +14086q 12 +20584q 13 +20710q 14 + 18212q 15 + 42532q 16 + 43108q 17 + 31870q 18 + 55516q 19 + · · · ),  ∞ X X  3 1 2 ΘF1 ⊕G3 ,ϕ44 (q) = x4 − (F1 ⊕ G3 ) q n = (−6q−12q 2 −18q 3 −144q 4 94 47 n=1 F1 ⊕G3 =n

−300q 5 − 216q 6 − 588q 7 − 1488q 8 − 1458q 9 − 1680q 10 − 3036q 11 − 3700q 12 − 4772q 13 −4936q 14 − 4812q 15 − 9320q 16 − 10392q 17 − 9068q 18 − 14904q 19 + · · · ),  ∞ X  X 3 ΘH4 ,ϕ11 (q) = x21 − H4 q n = 47 n=1 H4 =n

1 = (46q 2 +276q 4 +408q 6 −74q 7 −36q 8 −290q 9 +756q 10 −240q 11 +2488q 12 −1156q 13 47 +272q 14 − 4936q 15 + 568q 16 − 4188q 17 + 3518q 18 − 996q 19 + · · · ),  ∞ X  X 1 1 ΘH4 ,ϕ12 (q) = x1 x2 + H4 q n = (4q 2 +24q 4 +60q 6 −80q 7 +144q 8 −344q 9 +360q 10 188 47 n=1 H4 =n

11

12

−168q +388q +488q 13 −712q 14 −184q 15 −2084q 16 +960q 17 +404q 18 +2856q 19 +· · · ),  ∞ X X  1 3 2 ΘF3 ⊕H1 ,ϕ11 (q) = x1 − (F3 ⊕ H1 ) q n = (58q + 208q 2 + 204q 3 47 47 n=1 F3 ⊕H1 =n

+408q 4 + 1432q 5 + 1640q 6 + 1344q 7 + 2252q 8 + 1386q 9 + 2492q 10 + 6468q 11 + 4926q 12 +4780q 13 + 3578q 14 + 796q 15 + 9224q 16 + 8568q 17 + 3958q 18 + 11040q 19 + · · · ). Since the 19th determinant of the coefficients of the theta series is −502997951694710049247152046080 /506623120463, the 19 generalized theta series in the Theorem is a basis of S6 (Γ0 (47)).

THE DIRECT SUM OF 5 AND 6 BINARY QUADRATIC FORMS WITH DISCRIMINANT −47. 19

5. Representation Numbers of n Proposition 5.1. The difference between the following theta series of the quadratic forms [2.1] and the Eisenstein series [2.2] are linear combinations of the generalized theta series in the preceding theorem.The coefficients are given in the table [4]. Proof. Let’s see the situation in the case: 622680 3106344 2 135152 3 12815208 4 15408480 5 429680 6 ΘF6 ( q)−EF6 ( q) = q+ q + q + q + q + q 51911 51911 851 51911 51911 851 45598944 7 44425320 8 30532200 9 54985344 10 84001296 11 725048 12 + q + q + q + q + q + q 51911 51911 51911 51911 51911 851 24790992 13 80742600 14 2227312 15 101617080 16 62051952 17 146670576 18 q + q + q + q + q + q 51911 51911 851 51911 51911 51911 300453888 19 q + ··· + 51911 = c1 ΘF4 ,ϕ12 (q) + c2 ΘF4 ,ϕ22 (q) + c3 ΘG4 ,ϕ11 (q) + c4 ΘG4 ,ϕ12 (q) + c5 ΘF3 ⊕G1 ,ϕ12 (q)

+

+c6 ΘF3 ⊕G1 ,ϕ22 (q) + c7 ΘF3 ⊕G1 ,ϕ77 (q) + c8 ΘF3 ⊕G1 ,ϕ78 (q) + c9 ΘF2 ⊕G2 ,ϕ11 (q) +c10 ΘF2 ⊕G2 ,ϕ12 (q) + c11 ΘF2 ⊕G2 ,ϕ56 (q) + c12 ΘF2 ⊕G2 ,ϕ66 (q) + c13 ΘF1 ⊕G3 ,ϕ11 (q) +c14 ΘF1 ⊕G3 ,ϕ12 (q)+c15 ΘF1 ⊕G3 ,ϕ33 (q)+c16 ΘF1 ⊕G3 ,ϕ44 (q)+c17 ΘH4 ,ϕ11 (q)+c18 ΘH4 ,ϕ12 (q) +c19 ΘF3 ⊕H1 ,ϕ11 (q) By equating the coefficients of q n in both sides for n = 1, 2, 3, · · · , 19,we get an equation in coefficients. We repeat the same procedure for the other cases. At the end, by solving 19 linear equations in 19 variables we get the coefficients in table [4]. Similarly, we can do for the difference between the theta series of the other quadratic forms [2.1] and the Eisenstein series [2.2] . Corollary 5.2. The representation numbers for the quadratic forms [2.1](in these direct sums any form can be replaced by its inverse) are   ∞ X  ∞ X  X X 252 ∗ 1 1 n 2 σ (n) + c1 x1 x2 + F4 q + c2 x2 − F4 q n 51911 5 188 94 n=1 n=1 F4 =n

∞ X

F4 =n

F2 ⊕G2 =n

x21 −



∞ X

 2 1 G2 q n + c4 x1 x2 + G2 q n 47 188 n=1 G4 =n n=1 G2 =n     ∞ ∞ X X X X 1 2 n 2 +c5 x1 x2 + F3 ⊕ G1 q + c6 x7 − F3 ⊕ G1 q n 188 47 n=1 F3 ⊕G1 =n n=1 F3 ⊕G1 =n   ∞ X X X  X  1 3 +c7 x7 x8 + F3 ⊕ G1 q n +c8 x28 − F3 ⊕ G1 q n 188 94 n=1 F3 ⊕G1 =n F3 ⊕G1 =n F3 ⊕G1 =n     ∞ ∞ X X X X 6 1 2 n +c9 x1 − F2 ⊕ G2 q +c10 x1 x2 + F2 ⊕ G2 q n 47 188 n=1 F2 ⊕G2 =n n=1 F2 ⊕G2 =n   ∞ ∞ X X  X X  1 3 +c11 x5 x6 + F2 ⊕ G2 q n +c12 x26 − F2 ⊕ G2 q n 188 94 n=1 n=1 +c3

X 

X 

F2 ⊕G2 =n

˙ BARIS ¸ KENDIRL I˙

20 ∞ X

  ∞ X X  6 1 n +c13 − F1 ⊕ G3 q +c14 x1 x2 + F1 ⊕ G3 q n 47 188 n=1 F1 ⊕G3 =n n=1 F1 ⊕G3 =n   ∞ ∞ X X  X X  2 3 2 n 2 +c15 x3 − F1 ⊕ G3 q + c16 x4 − (F1 ⊕ G3 ) q n 47 94 n=1 F1 ⊕G3 =n n=1 F1 ⊕G3 =n   ∞ X  ∞ X  X X 3 1 +c17 x21 − H4 q n + c18 x1 x2 + H4 q n 47 188 n=1 H4 =n n=1 H4 =n   ∞ X X 6 c19 x21 − F3 ⊕ H1 q n . 47 n=1 X



x21

F3 ⊕H1 =n

Proof. The coefficients are the same coefficients with preceding Theorem. It follows from the preceding theorem.

References [1] Kendirli, B.:Cusp Forms in S4 (Γ0 (47)) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -47.International Journal of Mathematics and Mathematical Sciences Hindawi Publishing Corporation Volume 2012, Article ID 303492. [2] Kendirli, B.:Formulas for the Fourier coefficients of theta series for some quadratic forms.Turkish Journal of Mathematics(in press) [3] Kendirli, B.:Cusp Forms in S4 (Γ0 (79)) and the number of representations of positive integers by some direct sum of binary quadratic forms with discriminant -79. Bulletin of the Korean Mathematical Society Vol 49/3 2012. [4] https://sites.google.com/site/discriminantnegative47 [5] Ick Sun Eum, Dong Hwa Shin, and Dong Sung Yoon.: Representations by x21 + 2x22 + x23 + x24 + x1 x3 + x1 x4 + x2 x3 , arXiv:1102.5746v2 [math.NT] 6 Mar 2011 Department of Mathematics,Faculty of Arts and Sciences, Fatih University, Istanbul, Turkey E-mail address: [email protected]