3_Kokluce.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No. 4, 2012, 451-468 ISSN 1307-5543 – www.ejpam.com On the Number of Representation of Integers by the Direct Sum of BQFs with Discriminant −191 Bülent KÖKLÜCE Department of Mathematics, Faculty of Art and Sciences, Fatih University, Istanbul, Turkey Abstract. In this study we find a basis of the space S4(Γ0(191)) and derive explicit formulae for the number of representation of positive integers by all possible direct sum of 13 quadratic forms from the representatives x2 1 + x1 x2 + 48x2 2 , 2x2 1 + x1 x2 + 24x2 2 , 3x2 1 + x1 x2 + 16x2 2 , 4x2 1 + x1 x2 + 12x2 2 , 5x2 1 + 3x1 x2+ 10x2 2 , 6x2 1 + x1 x2+ 8x2 2 , 6x2 1 + 5x1 x2+ 9x2 2 of the class group of equivalence classes of quadratic forms with discriminant −191. 2010 Mathematics Subject Classifications: 11E20,11E25 Key Words and Phrases: Quadratic Forms, Representation Numbers, Theta Series,Cusp Forms 1. Introduction The problem of determining which positive integers are represented by quadratic forms has been studied extensively since earlier mathematicians. Fermat’s assertion of 1640 about representation of integers by the binary quadratic form x2 1+ x2 2 was proved by Euler. With La- grange’s four square theorem which states that, the quadratic form x2 1+ x2 2+ x2 3+ x2 4 represent all positive integers, the theory of universal quadratic forms has been started in 1770. It has been proved by Legendre in 1798 that the quadratic form x2 1 + x2 2 + x2 3 represent all positive integers except precisely the numbers of the form 4a(8k+ 7). Legendre also gives a general theory of binary quadratic forms in his study Theorie des Nombres in 1830. In 1930, Mordell [7] proved the five squares theorem, which states that the quadratic form x2 1+x2 2+x2 3+x2 4+x2 5 represent all positive definite binary quadratic forms. In 1997, Conway and Schneeberger proved that a positive definite integral quadratic form represents every positive integer if and only if it represents the integers 1,2,3,5,6,7,10,14, and 15. It is known as the 15 theorem and later has been proved by Bhargava [1] by a simpler method. Bhargava and Hanke [2] have shown in their studies that every integer valued quadratic form is universal if and only if it represent every integer less than 290. For a general information about the theory of quadratic forms one can see [6]. Email address: bkokluce@fatih.edu.tr http://www.ejpam.com 451 c© 2012 EJPAM All rights reserved. B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 452 Determination of positive integers represented by a given quadratic form Q is an inter- esting problem, but it is also interesting to ask in how many different ways is the integer n represented by Q? If we let r(n,Q) count the number of ways of representing n by Q, then we are asking for a description of the function r(n,Q). In these terms, the question of which integers n can be represented by Q means, for which n, r(n,Q) is > 0? Finding exact formulas for r(n,Q) is a classical problem in number theory. In some cases formulas can be obtained for r(n,Q), but such formulas are quite rare [3]. For instance, if we consider the quadratic form Q = x2 1 + x2 2 + x2 3 + x2 4 and n> 0, then we have the following Jacobi’s result [3]: r(n,Q) = 8 ∑ d\n 4∤d>0 d . Peterson [8], for the first time, considered the problem of representation of numbers by the direct sum of some binary quadratic forms. Kendirli [4] has given the number of repre- sentations of positive integers by some direct sum of binary quadratic forms with discriminant −79. In this study we obtain a basis of the space S4(Γ0(191)) and formulae for the number of representations of positive integers by some direct sums of binary quadratic forms with discriminant −191 which are all quadratic forms 8 variables. There exist 13 inequivalent classes of binary quadratic forms with discriminant −191. These are: F1 = x2 1 + x1 x2+ 48x2 2 Φ1 = 2x2 1 + x1 x2 + 24x2 2,Φ′1 = 2x2 1 − x1 x2+ 24x2 2 Ψ1 = 3x2 1 + x1 x2 + 16x2 2,Ψ′1 = 3x2 1 − x1 x2+ 16x2 2 Λ1 = 4x2 1 + x1 x2 + 12x2 2,Λ′1 = 4x2 1 − x1 x2+ 12x2 2, Υ1 = 5x2 1 + 3x1 x2+ 10x2 2,Υ′1 = 5x2 1 − 3x1 x2+ 10x2 2, Ω1 = 6x2 1 + x1 x2 + 8x2 2,Ω′1 = 6x2 1 − x1 x2+ 8x2 2, Π1 = 6x2 1 + 5x1 x2+ 9x2 2,Π′1 = 6x2 1 − 5x1 x2+ 9x2 2 , Here Φ′1, Ψ′1, Λ′1, Υ′1, Ω′1, and Π′1 are respectively the inverses of Φ1, Ψ1, Λ1, Υ1, Ω1, and Π1. Therefore the theta series of Φ′1, Ψ′1, Λ′1, Υ′1, Ω′1, and Π′1 are respectively same with the theta series of Φ1, Ψ1, Λ1, Υ1, Ω1, and Π1. Here F1 is the identity element and these quadratic forms form a group of order 13 which can be described as: Φ1,Φ2 1 = Λ1,Φ3 1 = Ω ′ 1,Φ4 1 = Ψ ′ 1,Φ5 1 = Π1,Φ6 1 = Υ ′ 1,Φ7 1 = Υ1,Φ8 1 = Π ′ 1,Φ9 1 = Ψ1, Φ10 1 = Ω1,Φ11 1 = Λ ′ 1,Φ12 1 = Φ ′ 1,Φ13 1 = F1. Since 191 is prime number then there is only one genus, i.e., the principal genus. For any quadratic form Q1 let Qk = Q1 + . . . + Q1 (k times) be kth direct sum of this quadratic form. In the present paper we obtain formulas r(n,Q) for any of the quadratic B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 453 forms Q = F4,Φ4,Ψ4,Λ4,Υ4,Ω4,Π4, Fi ⊕Φ j , Fi ⊕Ψ j, Fi ⊕Λ j , Fi ⊕Υ j , Fi ⊕Ω j, Fi ⊕Π j , Φi ⊕Ψ j,Φi ⊕Λ j,Φi ⊕Υ j ,Φi ⊕Ω j,Φi ⊕Π j ,Ψi ⊕Λ j ,Ψi ⊕Υ j,Ψi ⊕Ω j,Ψi ⊕Π j , Λi ⊕Υ j ,Λi ⊕Ω j,Λi ⊕Π j ,Υi ⊕Ω j ,Υi ⊕Π j ,Ωi ⊕Π j , Fi ⊕Φ j ⊕Ψl , Fi ⊕Φ j ⊕Λl , Fi ⊕Φ j ⊕Υl , Fi ⊕Φ j ⊕Ωl , Fi ⊕Φ j ⊕Πl ,Φi ⊕Ψ j ⊕Λl ,Φi ⊕Ψ j ⊕Υl ,Φi ⊕Ψ j ⊕Ωl , Φi ⊕Ψ j ⊕Πl ,Ψi ⊕Λ j ⊕Υl ,Ψi ⊕Λ j ⊕Ωl ,Ψi ⊕Λ j ⊕Ωl ,Ψi ⊕Λ j ⊕Πl ,Λi ⊕Υ j ⊕Ωl , Λi ⊕Υ j ⊕Πl ,Υi ⊕Ω j ⊕Πl , F1 ⊕Φ1 ⊕Ψ1 ⊕Λ1, F1 ⊕Φ1 ⊕Ψ1⊕Υ1, F1 ⊕Φ1 ⊕Ψ1 ⊕Ω1, F1 ⊕Φ1 ⊕Ψ1 ⊕Π1,Φ1 ⊕Ψ1 ⊕Λ1 ⊕Υ1,Φ1 ⊕Ψ1 ⊕Λ1 ⊕Ω1,Φ1 ⊕Ψ1⊕Λ1 ⊕Π1, Ψ1⊕Λ1 ⊕Υ1 ⊕Ω1,Ψ1 ⊕Λ1 ⊕Υ1 ⊕Π1, and Λ1 ⊕Υ1 ⊕Ω1⊕Π1 (1) (where i, j, l, m ≥ 1 and in any direct sum the sum of the indices is 4). In these direct sums one can replace the quadratic forms Φ1, Ψ1, Λ1, Υ1, Ω1, and Π1 by their inverses. 2. The Positive Definite Quadratic Forms In this section we give some definitions, an important theorem and evaluation of the quadratic forms. Definition 1. Let Q : Z2k→ Z be a positive definite integer-valued form of 2k variables, Q = 2k ∑ 1≤i≤ j≤2k bi j x i x j, bi j ∈ Z and the matrix A is defined by aii = 2bii, a ji = ai j = bi j for i < j. Let D be the discriminant of the quadratic form 2Q = 2k ∑ i, j=1 ai j x i x j i.e., the determinant of the matrix A. Let Ai j be the cofactors of ai j for 1 ≤ i ≤ j ≤ 2k. If δ = gcd(Aii 2 ,Ai j , for 1≤ i ≤ j ≤ 2k), then N := D δ is the smallest positive integer, called the level of Q, for which NA−1 is again an even integral matrix like A. ∆= (−1)kD is called the discriminant of the form Q. Theorem 1. Let Q : Z2k → Z be positive definite integer-valued form of 2k variables of level N and discriminant ∆. Then B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 454 1 The theta function ΘQ(q) = ∑ (n 1 ,n 2 ,...,nk)∈Z×Z×...×Z qQ(n1 ,n2,...,nk) = 1+ ∞ ∑ n=1 r(n;Q)qn,q = e2πiz is a modular form on Γ0(N) of weight k and character χd , i.e., ΘQ ∈ Mk(Γ0(N),χd), where χ∆(d) := � ∆ d � , d ∈ (Z/NZ)×, � ∆ d � is the Kronecker Character. 2 The homogeneous quadratic polynomials in 2k variables ϕi j = x i x j − 1 2k Ai j D 2Q, 1≤ i ≤ j ≤ 2k (2) are spherical functions of second order with respect to Q. 3 The theta series ΘQ,ϕi j (q) = ∞ ∑ n=1 ∑ Q=n ϕi j ! qn (3) is a cusp form in Sk+2(Γ0(N),χd). 4 If two quadratic forms Q1,Q2 have the same level N and the characteristic are χ1(d), χ2(d) respectively, then the direct sum Q1 ⊕Q2 of the quadratic forms has the same level N and the character χ1(d), χ2(d). Proof. See [7]. Now let’s look at the positive definite quadratic forms of discriminant −191. For the quadratic form F1 = x2 1 + x1 x2+ 48x2 2, 2F1 = 2x2 1 + 2x1x2 + 96x2 2 = � x1, x2 � � 2 1 1 96 �� x1 x2 � the determinant D = 191,A22 = 2, so δ = 1, N = D = 191 and the discriminant is ∆ = (−1)2/2191 = −191. Similarly, it can be easily seen that for any of the quadratic forms Φ1, Ψ1, Λ1, Υ1, Ω1 and Π1 the determinant, the discriminant and the character respectively are D = 191,∆= −191,χ(d) = � −191 d � . Consequently F1, Φ1, Ψ1, Λ1, Υ1, Ω1 and Π1 are quadratic forms whose theta series are in M1(Γ0(191), � −191 d � ). Hence by Theorem 1 F2, Φ2, Ψ2, Λ2, Υ2, Ω2, Π2, F1 ⊕ Φ1, F1 ⊕ Ψ1, F1 ⊕ Λ1, F1 ⊕ Υ1, F1 ⊕Ω1, F1 ⊕Π1, Φ1 ⊕Ψ1, Φ1 ⊕Λ1, Φ1 ⊕Υ1, Φ1 ⊕Ω1, Φ1 ⊕Π1, Ψ1 ⊕Λ1, Ψ1 ⊕Υ1, Ψ1 ⊕Ω1, B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 455 Ψ1 ⊕Π1, Λ1 ⊕Υ1, Λ1 ⊕Ω1, Λ1 ⊕Π1, Υ1 ⊕Ω1, Υ1 ⊕Π1, Ω1 ⊕Π1 are quadratic forms whose theta series are in M2(Γ0(191)). Theorem 2. Let Q be a positive definite quadratic form of 2k variables, k = 4,6,8, . . . whose theta series ΘQ is in Mk(Γ0(p)), p prime, then the Eisenstein part of ΘQ is E(q : Q) = 1+ ∞ ∑ n=1 (ασk−1(n)q n+βσk−1(n)q pn), where α= ik ρk pk/2− ik pk − 1 ,β = 1 ρk pk − ikpk/2 pk − 1 ,ρk = (−1)k/2 (k− 1)! (2π)k ζ(k). Proof. See [7]. We immediately obtain the following corollary. Corollary 1. Let Q be a positive definite quadratic form of 8 variables whose theta series ΘQ is in M4(Γ0(191)) then the Eisenstein part of ΘQ is E(q : Q) = 1+ ∞ ∑ n=1 (ασ3(n)q n+ βσ3(n)q 191n), where ρ4 = 3! (2π)4 ζ(4) = 3! (2π)4 . π4 90 = 1 240 , α= 240 1912− 1 1914− 1 = 240 1 1912+ 1 = 120 18241 β = 240 1914− 1912 1914− 1 = 240 1912 1912+ 1 = 1912 120 18241 and for any Q in (1) E(q : Q) = 1+ 120 18241 ∞ ∑ n=1 (qn+ 1912q191n)σ3(n) = 120 18241 ∞ ∑ n=1 σ∗3(n)q n where σ∗3(n) = ( σ3(n) if n≥ 1 and 191 ∤ n σ3(n) + 1912σ3(n/191) if 191 | n B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 456 3. The Selection of Spherical Functions Here we will select 47 spherical functions such that the corresponding generalized theta series span all the generalized theta series of (3) induced by spherical functions of the form (2). 1 For 2F2 =2x2 1 + 2x1 x2+ 96x2 2 + 2x2 3 + 2x3x4 + 96x2 4 = � x1, x2, x3, x4 �      2 1 0 0 1 96 0 0 0 0 2 1 0 0 1 96           x1 x2 x3 x4      the determinant D = 1912,A11 = 76.191. By putting 2k = 4, Q = F2, and appropriate i, j in Theorem 1, we get the spherical function of second order with respect to F2 as: ϕ12 = x2 1 − 1 4 96.191 1912 2F2 = x2 1 − 48 191 F2, 2 For 2Φ2 = 4x2 1+2x1 x2+48x2 2+4x2 3+2x3 x4+48x2 4, by taking A11 = 48.191, A12 = −191 we get ϕ11 = x2 11 − 1 4 48.191 1912 2Φ2 = x2 11− 24 191 Φ2, ϕ12 = x1 x2+ 1 4 191 1912 2Φ2 = x1 x2 + 1 2.191 Φ2, which will be spherical functions of second order with respect to Φ2. 3 For 2Ψ2 = 6x2 1+2x1 x2+32x2 2+6x2 3+2x3 x4+32x2 4, by taking A22 = 6.191, A33 = 32.191 we get; ϕ22 = x2 2 − 3 191 Ψ2,ϕ33 = x2 3 − 16 191 Ψ2, which will be spherical functions of second order with respect to Ψ2. 4 For 2Λ2 = 8x2 1+2x1 x2+24x2 2+8x2 3+2x3 x4+24x2 4, by taking A11 = 24.191, A12 = −191 we get; ϕ11 = x2 1 − 12 191 Λ2,ϕ12 = x1 x2 + 1 2.191 Λ2, which will be spherical functions of second order with respect to Λ2. 5 For 2Υ2 = 10x2 1 + 6x1 x2+ 20x2 2 + 10x2 3 + 6x3 x4+ 20x2 4 by taking A11 = 20.191, A22 = 10.191 we have; ϕ11 = x2 1 − 10 191 Υ2,ϕ22 = x2 2 − 5 191 Υ2, which will be spherical functions of second order with respect to Λ2. B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 457 6 For 2Ω2 = 12x2 1 + 2x1 x2+ 16x2 2 + 12x2 3 + 2x3 x4+ 16x2 4, by taking A12 = −191, A22 = 12.191 we get; ϕ12 = x1 x2+ 1 2.191 Ω2,ϕ22 = x2 2 − 6 191 Ω2, which will be spherical functions of second order with respect to Ω2. 7 For 2Π2 = 12x2 1 + 10x1x2 + 18x2 2 + 12x2 3 + 10x3x4 + 18x2 4, by taking A11 = 18.191, A22 = 12.191 we get, ϕ11 = x2 1 − 9 191 Π2,ϕ22 = x2 2 − 6 191 Π2, which will be spherical functions of second order with respect to Π2. 8 For 2(F1⊕Φ1) = 2x2 1 +2x1x2+96x2 2+4x2 3+2x3 x4+48x2 4 the determinant D = 1912, A12 = −191, A33 = 48.191, the spherical functions of second order with respect to F1 ⊕Φ1 are; ϕ12 = x1 x2+ 1 2.191 (F1 ⊕Φ1),ϕ33 = x2 3 − 24 191 (F1 ⊕Φ1), 9 For 2(F1⊕Ψ1) = 2x2 1+2x1x2+96x2 2+6x2 3+2x3x4+32x2 4 the determinant D = 1912, A22 = 2.191, A34 = −191, the spherical functions of second order with respect to F1⊕Ψ1 are; ϕ22 = x2 2 − 1 191 (F1 ⊕Ψ1),ϕ34 = x3 x4 + 1 2.191 (F1 ⊕Ψ1), 10 For 2(F1⊕Λ1) = 2x2 1 +2x1x2+96x2 2+8x2 3+2x3 x4+24x2 4 the determinant D = 1912, A12 =−191, A44 = 8.191, the spherical functions of second order with respect to F1⊕Λ1 are; ϕ12 = x1 x2+ 1 2.191 (F1 ⊕Λ1),ϕ44 = x2 4 − 4 191 (F1 ⊕Λ1), 11 For 2(F1⊕Υ1) = 2x2 1+2x1x2+96x2 2+10x2 3+6x3 x4+20x2 4 the determinant D = 1912, A11 = 96.191, A34 = −3.191, the spherical functions of second order with respect to F1 ⊕Υ1 are; ϕ11 = x2 1 − 48 191 (F1 ⊕Υ1),ϕ34 = x3 x4 + 3 2.191 (F1 ⊕Υ1), 12 For 2(F1⊕Ω1) = 2x2 1+2x1x2+96x2 2+12x2 3+2x3x4+16x2 4 the determinant D = 1912, A12 = −191, A33 = 20.191, the spherical functions of second order with respect to F1 ⊕Ω1 are; ϕ12 = x1 x2+ 1 2.191 (F1 ⊕Ω1),ϕ33 = x2 3 − 10 191 (F1 ⊕Ω1), B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 458 13 For 2(F1⊕Π1) = 2x2 1+2x1x2+96x2 2+12x2 3+10x3x4+18x2 4 the determinant D = 1912, A22 = 2.191, A34 = −5.191, the spherical functions of second order with respect to F1 ⊕Π1 are; ϕ22 = x2 2 − 1 191 (F1 ⊕Π1),ϕ34 = x3 x4 + 5 2.191 (F1 ⊕Π1), 14 For 2(Φ1⊕Ψ1) = 4x2 1+2x1 x2+48x2 2+6x2 3+2x3 x4+32x2 4 the determinant D = 1912, A11 = 48.191, A22 = 4.191, the spherical functions of second order with respect to Φ1 ⊕Ψ1 are; ϕ11 = x2 1 − 24 191 (Φ1 ⊕Ψ1),ϕ22 = x2 2 − 2 191 (Φ1 ⊕Ψ1), 15 For 2(Φ1 ⊕ Λ1) = 4x2 1 + 2x1 x2 + 48x2 2 + 8x2 3 + 2x3 x4 + 24x2 4, D = 1912, A12 = −191, A33 = 24.191, the spherical functions of second order with respect to Φ1 +Λ1 are; ϕ12 = x1 x2+ 1 2.191 (Φ1 ⊕Λ1),ϕ33 = x2 3 − 12 191 (Φ1 ⊕Λ1), 16 For 2(Φ1⊕Υ1) = 4x2 1 + 2x1 x2+ 48x2 2 + 10x2 3 + 6x3 x4+ 20x2 4, D = 1912, A12 = −191, A22 = 4.191, the spherical functions of second order with respect to Φ1 +Υ1 are; ϕ12 = x1 x2 + 1 2.191 (Φ1 ⊕Υ1),ϕ22 = x2 2 − 2 191 (Φ1 ⊕Υ1). 17 For 2(Φ1⊕Ω1) = 4x2 1 +2x1x2+48x2 2+12x2 3+2x3x4+16x2 4, D = 1912, A33 = 16.191, A34 =−191, the spherical functions of second order with respect to Φ1 ⊕Ω1 are; ϕ33 = x2 3 − 8 191 (Φ1 ⊕Ω1),ϕ34 = x3 x4 + 1 2.191 (Φ1 ⊕Ω1). 18 For 2(Φ1⊕Π1) = 4x2 1+2x1x2+48x2 2+12x2 3+10x3x4+18x2 4, D = 1912, A11 = 48.191, A33 = 18.191, the spherical functions of second order with respect to Φ1 ⊕Π1 are; ϕ11 = x2 1 − 24 191 (Φ1⊕Π1),ϕ33 = x2 3 − 9 191 (Φ1 ⊕Π1). 19 For 2(Ψ1 ⊕Λ1) = 6x2 1 + 2x1 x2 + 32x2 2 + 8x2 3 + 2x3 x4 + 24x2 4, D = 1912, A12 = −191, A22 = 6.191, the spherical functions of second order with respect to Ψ1⊕Λ1 are; ϕ12 = x1 x2 + 1 2.191 (Ψ1⊕Λ1),ϕ22 = x2 2 − 3 191 (Ψ1⊕Λ1), 20 For 2(Ψ1⊕Υ1) = 6x2 1+2x1x2+32x2 2+10x2 3+6x3 x4+20x2 4, D = 1912, A33 = 20.191, A44 = 10.191, the spherical functions of second order with respect to Ψ1 ⊕Υ1 are; ϕ33 = x2 3 − 10 191 (Ψ1⊕Υ1),ϕ44 = x2 4 − 5 191 (Ψ1⊕Υ1). B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 459 21 For 2(Ψ1⊕Ω1) = 6x2 1 + 2x1x2 + 32x2 2 + 12x2 3 + 2x3 x4 + 16x2 4, D = 1912, A12 = −191, A33 = 16.191, the spherical functions of second order with respect to Ψ1 ⊕Ω1 are; ϕ12 = x1 x2+ 1 2.191 (Ψ1 ⊕Ω1),ϕ33 = x2 3 − 8 191 (Ψ1⊕Ω1). 22 For 2(Ψ1⊕Π1) = 6x2 1+2x1 x2+32x2 2+12x2 3+10x3x4+18x2 4, D = 1912, A11 = 32.191, A34 =−5.191, the spherical functions of second order with respect to Ψ1 ⊕Π1 are; ϕ11 = x2 1 − 16 191 (Ψ1⊕Π1),ϕ34 = x3 x4 + 5 2.191 (Ψ1⊕Π1). 23 For 2(Λ1⊕Υ1) = 8x2 1+2x1x2+24x2 2+10x2 3+6x3x4+20x2 4, D = 1912, A11 = 24.191, A22 = 8.191, the spherical functions of second order with respect to Λ1 ⊕Υ1 are; ϕ11 = x2 1 − 12 191 (Λ1 ⊕Υ1),ϕ22 = x2 2 − 4 191 (Λ1 ⊕Υ1). 24 For 2(Λ1+Ω1) = 8x2 1 +2x1x2+24x2 2+12x2 3+2x3x4+16x2 4, D = 1912, A33 = 16.191, A44 = 12.191, the spherical functions of second order with respect to Λ1 ⊕Ω1 are; ϕ33 = x2 3 − 8 191 (Λ1 ⊕Ω1),ϕ44 = x2 4 − 6 191 (Λ1 ⊕Ω1). 4. The Solutions of Q = n and the Theta Series Associated to the Quadratic Forms The equation F1 = x2 1 + x1 x2+ 48x2 2 = n have the following solutions: n = 1⇒ x1 = ±1, x2 = 0 n = 4⇒ x1 = ±4, x2 = 0 n = 9⇒ x1 = ±3, x2 = 0 n = 16⇒ x1 = ±4, x2 = 0 n = 25⇒ x1 = ±5, x2 = 0 n = 36⇒ x1 = ±6, x2 = 0 and there is no integral solutions for: n = 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47. Thus the theta series of F1 is given by ΘF1 (q) = 1+ 2q+ 2q4+ 2q9+ 2q16 + 2q25 + 2q36+ . . . By a similar way theta series of Φ1, Ψ1, Λ1, Υ1, Ω1, and Π1 are obtained as follows: ΘΦ1 (q) = 1+ 2q2+ 2q8+ 2q18 + 2q24 + 2q25 + 2q27+ 2q30 + 2q32 + 2q34 + 2q39 + 2q45+ . . . B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 460 ΘΨ1 (q) = 1+ 2q3 + 2q12+ 2q16 + 2q18+ 2q20 + 2q26 + 2q27 + 2q30 + 2q40+ 2q46 + . . . ΘΛ1 (q) = 1+ 2q4 + 2q12+ 2q15 + 2q16+ 2q17 + 2q26 + 2q30+ 2q36 + 2q45+ . . . ΘΥ1 (q) = 1+ 2q5+ 2q10 + 2q12+ 2q18 + 2q20 + 2q24+ 2q36 + 2q39+ 2q40 + 2q45 + 2q46 + . . . ΘΩ1 (q) = 1+ 2q6 + 2q8 + 2q13+ 2q15 + 2q24+ 2q30 + 2q32 + 2q34+ 2q36 + 2q40+ . . . ΘΠ1 (q) = 1+ 2q6+ 2q9 + 2q10 + 2q20 + 2q23+ 2q24 + 2q32 + 2q36 + 2q40 + 2q43+ . . . Here as an example we will compute the theta series of F4. Theta series ΘQ(q) for any quadratic form in (1) are obtained in a similar way. ΘF4 (q) =ΘF1 (q).ΘF1 (q).ΘF1 (q) ·ΘF1 (q) = 1+ 8q+ 24q2+ 32q3+ 24q4+ 48q5+ 96q6 + 64q7+ 24q8+ 104q9+ 144q10+ 96q11+ 96q12+ 112q13+ 192q14+ 192q15 + 24q16 + 144q17+ 312q18+ 160q19+ 144q20+ 256q21+ 288q22+ 192q23 + 96q24 + 248q25+ 336q26+ 320q27+ 192q28+ 240q29+ 576q30+ 256q31 + 24q32 + 384q33+ 432q34+ 384q35+ 312q36+ 304q37+ 480q38+ 448q39 + 144q40+ 336q41+ 768q42+ 352q43+ 288q44+ 624q45+ 576q46+ 384q47+ . . . Theorem 3. The following system of generalized fourfold theta-series is a basis of S4(Γ0(191)), ΘF2,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ F2=n (191x2 1 − 48F2)q n, ΘΦ2,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ Φ2=n (191x2 1 − 24Φ2)q n, ΘΦ2,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ Φ2=n (191x1x2 + 1 2 Φ2)q n, ΘΨ2,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ2=n (191x2 2 − 3Ψ2)q n, ΘΨ2,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ2=n (191x2 3 − 16Ψ2)q n, ΘΛ2,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ Λ2=n (191x2 1 − 12Λ2)q n, ΘΛ2,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ Λ2=n (191x1x2 + 1 2 Λ2)q n, ΘΥ2,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ Υ2=n (191x2 1 − 10Υ2)q n, ΘΥ2,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ Υ2=n (191x2 2 − 5Υ2)q n, B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 461 ΘΩ2,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ Ω2=n (191x1x2+ 1 2 Ω2)q n, ΘΩ2,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ Ω2=n (191x2 2 − 6Ω2)q n, ΘΠ2,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ Π2=n (191x2 1 − 9Π2)q n, ΘΠ2,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ Π2=n (191x2 2 − 6Π2)q n, ΘF1⊕Φ1,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Φ1=n (191x1x2 + 1 2 (F1 ⊕Φ1))q n, ΘF1⊕Φ1,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Φ1=n (191x2 3 − 24(F1⊕Φ1))q n, ΘF1⊕Ψ1,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Ψ1=n (191x2 2 − (F1 ⊕Ψ1))q n, ΘF1⊕Ψ1,ϕ34 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Ψ1=n (191x3x4+ 1 2 (F1 ⊕Ψ1))q n, ΘF1⊕Λ1,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Λ1=n (191x1x2 + 1 2 (F1 ⊕Λ1))q n, ΘF1⊕Λ1,ϕ44 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Λ1=n (191x2 4 − 4(F1 ⊕Λ1))q n, ΘF1⊕Υ1,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Υ1=n (191x2 1 − 48(F1⊕Υ1))q n, ΘF1⊕Υ1,ϕ34 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Υ1=n (191x3x4+ 3 2 (F1 ⊕Υ1))q n, ΘF1⊕Ω1,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Ω1=n (191x1x2+ 1 2 (F1 ⊕Ω1))q n, ΘF1⊕Ω1,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Ω1=n (191x2 3 − 10(F1⊕Ω1))q n, ΘF1⊕Π1,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Π1=n (191x2 2 − (F1 ⊕Π1))q n, B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 462 ΘF1⊕Π1 ,ϕ34 (q) = 1 191 ∞ ∑ n=1 ∑ F1⊕Π1=n (191x3x4 + 5 2 (F1 ⊕Π1))q n, ΘΦ1⊕Ψ1,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Ψ1=n (191x2 1 − 24(Φ1⊕Ψ1))q n, ΘΦ1⊕Ψ1,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Ψ1=n (191x2 2 − 2(Φ1⊕Ψ1))q n, ΘΦ1⊕Λ1,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Λ1=n (191x1x2 + 1 2 (Φ1 ⊕Λ1))q n, ΘΦ1⊕Λ1,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Λ1=n (191x2 3 − 12(Φ1⊕Λ1))q n, ΘΦ1⊕Υ1,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Υ1=n (191x1x2+ 1 2 (Φ1⊕Υ1))q n, ΘΦ1⊕Υ1,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Υ1=n (191x2 2 − 2(Φ1⊕Υ1))q n, ΘΦ1⊕Ω1,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Ω1=n (191x2 3 − 8(Φ1⊕Ω1))q n, ΘΦ1⊕Ω1,ϕ34 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Ω1=n (191x3x4+ 1 2 (Φ1 ⊕Ω1))q n, ΘΦ1⊕Π1,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Π1=n (191x2 1 − 24(Φ1⊕Π1))q n, ΘΦ1⊕Π1 ,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ Φ1⊕Π1=n (191x2 3 − 9(Φ1 ⊕Π1))q n, ΘΨ1⊕Λ1,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ1⊕Λ1=n (191x1x2+ 1 2 (Ψ1⊕Λ1))q n, ΘΨ1⊕Λ1,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ1⊕Λ1=n (191x2 2 − 3(Ψ1⊕Λ1))q n, ΘΨ1⊕Υ1,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ1⊕Υ1=n (191x2 3 − 10(Ψ1⊕Υ1))q n, ΘΨ1⊕Υ1,ϕ44 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ1⊕Υ1=n (191x2 4 − 5(Ψ1⊕Υ1))q n, B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 463 ΘΨ1⊕Ω1,ϕ12 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ1⊕Ω1=n (191x1x2 + 1 2 (Ψ1⊕Ω1))q n, ΘΨ1⊕Ω1,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ1⊕Ω1=n (191x2 3 − 8(Ψ1⊕Ω1))q n, ΘΨ1⊕Π1 ,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ1⊕Π1=n (191x2 1 − 16(Ψ1⊕Π1))q n, ΘΨ1⊕Π1 ,ϕ34 (q) = 1 191 ∞ ∑ n=1 ∑ Ψ1⊕Π1=n (191x3x4+ 5 2 (Ψ1⊕Π1))q n, ΘΛ1⊕Υ1,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ Λ1⊕Υ1=n (191x2 1 − 12(Λ1⊕Υ1))q n, ΘΛ1⊕Υ1,ϕ22 (q) = 1 191 ∞ ∑ n=1 ∑ Λ1⊕Υ1=n (191x2 2 − 4(Λ1⊕Υ1))q n, ΘΛ1⊕Ω1,ϕ33 (q) = 1 191 ∞ ∑ n=1 ∑ Λ1⊕Ω1=n (191x2 3 − 8(Λ1⊕Ω1))q n, ΘΛ1⊕Ω1,ϕ44 (q) = 1 191 ∞ ∑ n=1 ∑ Λ1⊕Ω1=n (191x2 4 − 6(Λ1⊕Ω1))q n. Proof. F2 = x2 1 + x1 x2 + 48x2 2 + x2 3 + x3 x4+ 48x2 4 = n has the following solutions; n= 1⇒ the solutions are; (±1,0,0,0), (0,0,±1,0), n= 2⇒ the solutions are;(±1,0,±1,0), n= 4⇒ the solutions are; (±2,0,0,0), (0,0,±2,0), n= 5⇒ the solutions are;(±2,0,±1,0), (±1,0,±2,0), n= 8⇒ the solutions are;(±2,0,±2,0), n= 9⇒ the solutions are;(±3,0,0,0), (0,0,±3,0), n= 10⇒ the solutions are;(±3,0,±1,0), (±1,0,±3,0), n= 13⇒ the solutions are;(±3,0,±2,0), (±2,0,±3,0), n= 16⇒ the solutions are;(±4,0,0,0), (0,0,±4,0), n= 17⇒ the solutions are;(±4,0,±1,0), (±1,0,±4,0), n= 18⇒ the solutions are;(±3,0,±3,0), n= 20⇒ the solutions are;(±4,0,±2,0), (±2,0,±4,0), n= 25⇒ the solutions are;(±5,0,0,0), (0,0,±5,0), (±4,0,±3,0), (±3,0,±4,0), n= 26⇒ the solutions are;(±5,0,±1,0), (±1,0,±5,0), n= 29⇒ the solutions are;(±5,0,±2,0), (±2,0,±5,0), n= 32⇒ the solutions are;(±4,0,±4,0), n= 34⇒ the solutions are;(±5,0,±3,0), (±3,0,±5,0), n= 36⇒ the solutions are;(±6,0,0,0), (0,0,±6,0), n= 37⇒ the solutions are;(±6,0,±1,0), (±1,0,±6,0), B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 464 n= 40⇒ the solutions are;(±6,0,±2,0), (±2,0,±6,0), n= 41⇒ the solutions are;(±5,0,±4,0), (±4,0,±5,0), n= 45⇒ the solutions are;(±6,0,±3,0), (±3,0,±6,0), and for n =3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 38, 39, 42, 43, 44, 46 there is no integral solution. Hence; ΘF2,ϕ11 (q) = 1 191 ∞ ∑ n=1 ∑ F2=n (191x2 1 − 48F2)q n = 1 191 ((191.2− 48.4)q+ (191.1.4− 48.4.2)q2+ (191.4.2− 48.4.4)q4 + (191.4.4+ 191.1.4− 48.8.5)q5+ (191.4.4− 48.4.8)q8+ (191.9.2− 48.4.9)q9 + (191.9.4+ 191.1.4− 48.8.10)q10+ (191.9.4+ 191.4.4− 48.8.13)q13 + (191.16.2− 48.4.16)q16+ (191.16.4+ 191.1.4− 48.8.17)q17 + (191.9.4− 48.4.18)q18+ (191.16.4+ 191.4.4− 48.8.20)q20 + (191.25.2+ 191.16.4+ 191.9.4− 48.12.25)q25 + (191.25.4+ 191.1.4− 48.8.26)q26 + (191.25.4+ 191.4.4− 48.8.29)q29+ (191.16.4− 48.4.32)q32 + (191.25.4+ 191.9.4− 48.8.34)q34+ (191.36.2− 48.4.36)q36 + (191.36.4+ 191.1.4− 48.8.37)q37+ (191.36.4+ 191.4.4− 48.8.40)q40 + (191.25.4+ 191.16.4− 48.8.41)q41+ (191.36.4+ 191.9.4− 48.8.45)q45+ . . .) Therefore, ΘF2,ϕ11 (q) = 1 191 (190q+ 380q2+ 760q4+ 1900q5+ 1520q8+ 1710q9+ 3800q10 + 4940q13+ 3040q16+ 6460q17+ 3420q18+ 7600q20+ 14250q25+ 9880q26 + 11020q29+ 6080q32+ 12920q34+ 6840q36+ 14060q37+ 15200q40 + 15580q41+ 17100q45+ . . .) is obtained. We obtained the remaining theta series by similar calculations. For a complete list of theta series see Table 1 in [5]. The Calculations in this article are done by using the software packages Pari GP and Maple. The 47−th determinant of the coefficients of Theta Series is −54152562377765212169769805340948504021762461314755516203054592423781 22484013765646204060917570748490727120039404251791740314128696213504 00000000 1 19147 6= 0. So, the Theta series in Theorem 3 is a basis of S4(Γ0(191)). B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 465 5. Representation Numbers of n Proposition 1. The differences between the Theta series of the quadratic forms in (1) (in these direct sums any form can be replaced by its inverse) and the Eisenstein Series E(q : Q) = 1+ 120 18241 ∞ ∑ n=1 (qn+ 1912q191n)σ3(n) = 120 18241 ∞ ∑ n=1 σ∗3(n)q n = 1+ 120 18241 q+ 120.9 18241 q2 + 120.28 18241 q3 + 120.73 18241 q4 + 120.126 18241 q5 + 120.252 18241 q6 + 120.344 18241 q7 +120.585 18241 q8 + 120.757 18241 q9 + 120.1134 18241 q10 + 120.1332 18241 q11 + 120.2044 18241 q12 + 120.2198 18241 q13 + +120.3096 18241 q14 + 120.3528 18241 q15 + 120.4681 18241 q16 + 120.4914 18241 q17 + 120.6813 18241 q18 + 120.6860 18241 q19 +120.9198 18241 q20 + 120.9632 18241 q21 + 120.11988 18241 q22 + 120.12168 18241 q23 + 120.16380 18241 q24 +120.15751 18241 q25 + 120.19782 18241 q26 + 120.20440 18241 q27 + 120.25112 18241 q28 + 120.24390 18241 q29 +120.31752 18241 q30 + 120.29792 18241 q31 + 120.37449 18241 q32 + 120.37296 18241 q33 + 120.44226 18241 q34 +120.43344 18241 q35 + 120.55261 18241 q36 + 120.50654 18241 q37 + 120.61740 18241 q38 + 120.61544 18241 q39 +120.73710 18241 q40 + 120.68922 18241 q41 + 120.86688 18241 q42 + 120.79508 18241 q43 + 120.97236 18241 q44 +120.95382 18241 q45 + 120.109512 18241 q46 + 120.103824 18241 q47 + . . . are linear combination of the Theta Series in the preceding theorem. Proof. Now we will consider the case; ΘF4 − E(q : F4) =c1ΘF2,ϕ11 (q) + c2ΘΦ2,ϕ11 (q) + c3ΘΦ2,ϕ12 (q) + c4ΘΨ2,ϕ22 (q) + c5ΘΨ2,ϕ33 (q)+ c6ΘΛ2,ϕ11 (q)+ c7ΘΛ2,ϕ12 (q)+ c8ΘΥ2,ϕ11 (q) + c9ΘΥ2,ϕ22 (q) + c10ΘΩ2,ϕ12 (q)+ c11ΘΩ2,ϕ22 (q)+ c12ΘΠ2,ϕ11 (q) + c13ΘΠ2,ϕ22 (q) + c14ΘF1⊕Φ1,ϕ12 (q) + c15ΘF1⊕Ψ1 ,ϕ34 (q)+ c16ΘF1⊕Ψ1,ϕ22 (q) + c17ΘF1⊕Ψ1 ,ϕ34 (q) + c18ΘF1⊕Λ1,ϕ12 (q) + c19ΘF1⊕Λ1,ϕ44 (q) + c20ΘF1⊕Υ1,ϕ11 (q) + c21ΘF1⊕Υ1,ϕ34 (q) + c22ΘF1⊕Ω1,ϕ12 (q) + c23ΘF1⊕Ω1,ϕ33 (q) + c24ΘF1⊕Π1 ,ϕ22 (q)+ c25ΘF1⊕Π1,ϕ34 (q) + c26ΘΦ1⊕Ψ1 ,ϕ11 (q)+ c27ΘΦ1⊕Ψ1,ϕ22 (q) + c28ΘΦ1⊕Λ1,ϕ12 (q)+ c29ΘΦ1⊕Λ1,ϕ33 (q) + c30ΘΦ1⊕Υ1 ,ϕ12 (q)+ c31ΘΦ1⊕Υ1,ϕ22 (q) + c32ΘΦ1⊕Ω1,ϕ33 (q) + c33ΘΦ1⊕Ω1,ϕ34 (q) + c34ΘΦ1⊕Π1 ,ϕ11 (q) + c35ΘΦ1⊕Π1 ,ϕ33 (q)+ c36ΘΨ1⊕Λ1,ϕ12 (q)+ c37ΘΨ1⊕Λ1,ϕ22 (q) + c38ΘΨ1⊕Υ1,ϕ33 (q) + c39ΘΨ1⊕Υ1,ϕ44 (q) + c40ΘΨ1⊕Ω1,ϕ12 (q)+ c41ΘΨ1⊕Ω1,ϕ33 (q) + c42ΘΨ1⊕Π1,ϕ11 (q) + c43ΘΨ1⊕Π1,ϕ34 (q)+ c44ΘΛ1⊕Υ1,ϕ11 (q) + c45ΘΛ1⊕Υ1,ϕ22 (q) + c46ΘΛ1⊕Ω1,ϕ33 (q) + c47ΘΛ1⊕Ω1,ϕ44 (q) =(145808/18241)q+ (436704/18241)q2+ (580352/18241)q3 + (429024/18241)q4+ (860448/18241)q5+ (1720896/18241)q6 + (1126144/18241)q7+ (367584/18241)q8+ (1806224/18241)q9 B. KÖKLÜCE / Eur. J. Pure Appl. Math, 5 (2012), 451-468 466 + (2490624/18241)q10+ (43008/493)q11+ (1505856/18241)q12 + (1779232/18241)q13+ (3130752/18241)q14+ (3078912/18241)q15 − (123936/18241)q16+ (2037024/18241)q17+ (4873632/18241)q18 + (2095360/18241)q19+ (1522944/18241)q20+ (3513856/18241)q21 + (103104/493)q22+ (2042112/18241)q23− (214464/18241)q24 + (2633648/18241)q25+ (3755136/18241)q26+ (3384320/18241)q27 + (488832/18241)q28+ (1451040/18241)q29+ (6696576/18241)q30 + (1094656/18241)q31− (4056096/18241)q32+ (68352/493)q33 + (2572992/18241)q34+ (1803264/18241)q35− (940128/18241)q36 − (533216/18241)q37+ (1346880/18241)q38+ (786688/18241)q39 − (6218496/18241)q40− (2141664/18241)q41+ (3606528/18241)q42 − (3120128/18241)q43− (173376/493)q44− (63456/18241)q45 − (2634624/18241)q46− (5454336/18241)q47+ . . . . By equating the coefficients of qn in both sides for n = 1,2,3, . . . , 47, we get an equation in coefficients ci for i = 1, . . . 47. For the list of coefficients of any form in (1) see Table 2 in [5]. Corollary 2. The representation numbers r(n, F4) are ΘF4 − E(q : F4) = 120 18241 σ∗3(n)+ 1 191 (c1 ∑ F2=n (191x2 1 − 48F2) + c2 ∑ Φ2=n (191x2 1 − 24Φ2) + c3 ∑ Φ2=n (191x1x2 + 1 2 Φ2) + c4 ∑ Ψ2=n (191x2 2 − 3Ψ2) + c5 ∑ Ψ2=n (191x2 3 − 16Ψ2) + c6 ∑ Λ2=n (191x2 1 − 12Λ2) + c7 ∑ Λ2=n (191x1x2 + 1 2 Λ2) + c8 ∑ Υ2=n (191x2 1 − 10Υ2) + c9 ∑ Υ2=n (191x2 2 − 5Υ2) + c10 ∑ Ω2=n (191x1x2+ 1 2 Ω2) + c11 ∑ Ω2=n (191x2 2 − 6Ω2) + c12 ∑ Π2=n (191x2 1 − 9Π2) + c13 ∑ Π2=n (191x2 2 − 6Π2) + c14 ∑ F1⊕Φ1=n (191x1x2 + 1 2 (F1 ⊕Φ1)) + c15 ∑ F1⊕Φ1=n (191x2 3 − 24(F1⊕Φ1)) + c16 ∑ F1⊕Ψ1=n (191x2 2 − (F1 ⊕Ψ1)) + c17 ∑ F1⊕Ψ1=n (191x3x4+ 1 2 (F1 ⊕Ψ1)) + c18 ∑ F1⊕Λ1=n (191x1x2 + 1 2 (F1 ⊕Λ1)) + c19 ∑ F1⊕Λ1=n (191x2 4 − 4(F1 ⊕Λ1)) REFERENCES 467 + c20 ∑ F1⊕Υ1=n (191x2 1 − 48(F1 ⊕Υ1)) + c21 ∑ F1⊕Υ1=n (191x3x4+ 3 2 (F1 ⊕Υ1)) + c22 ∑ F1⊕Ω1=n (191x1x2+ 1 2 (F1 ⊕Ω1)) + c23 ∑ F1⊕Ω1=n (191x2 3 − 10(F1 ⊕Ω1)) + c24 ∑ F1⊕Π1=n (191x2 2 − (F1 ⊕Π1))+ c25 ∑ F1⊕Π1=n (191x3x4 + 5 2 (F1 ⊕Π1)) + c26 ∑ Φ1⊕Ψ1=n (191x2 1 − 24(Φ1⊕Ψ1))+ c27 ∑ Φ1⊕Ψ1=n (191x2 2 − 2(Φ1⊕Ψ1)) + c28 ∑ Φ1⊕Λ1=n (191x1x2 + 1 2 (Φ1 ⊕Λ1))+ c29 ∑ Φ1⊕Λ1=n (191x2 3 − 12(Φ1⊕Λ1)) + c30 ∑ Φ1⊕Υ1=n (191x1x2+ 1 2 (Φ1 ⊕Υ1)) + c31 ∑ Φ1⊕Υ1=n (191x2 2 − 2(Φ1⊕Υ1)) + c32 ∑ Φ1⊕Ω1=n (191x2 3 − 8(Φ1⊕Ω1)))+ c33 ∑ Φ1⊕Ω1=n (191x3x4 + 1 2 (Φ1 ⊕Ω1) + c34 ∑ Φ1⊕Π1=n (191x2 1 − 24(Φ1⊕Π1)) + c35 ∑ Φ1⊕Π1=n (191x2 3 − 9(Φ1⊕Π1)) + c36 ∑ Ψ1⊕Λ1=n (191x1x2+ 1 2 (Ψ1 ⊕Λ1))+ c37 ∑ Ψ1⊕Λ1=n (191x2 2 − 3(Ψ1⊕Λ1)) + c38 ∑ Ψ1⊕Υ1=n (191x2 3 − 10(Ψ1⊕Υ1)) + c39 ∑ Ψ1⊕Υ1=n (191x2 4 − 5(Ψ1⊕Υ1)) + c40 ∑ Ψ1⊕Ω1=n (191x1x2+ 1 2 (Ψ1 ⊕Ω1)) + c41 ∑ Ψ1⊕Ω1=n (191x2 3 − 8(Ψ1⊕Ω1)) + c42 ∑ Ψ1⊕Π1=n (191x2 1 − 16(Ψ1⊕Π1)) + c43 ∑ Ψ1⊕Π1=n (191x3x4+ 5 2 (Ψ1 ⊕Π1)) + c44 ∑ Λ1⊕Υ1=n (191x2 1 − 12(Λ1⊕Υ1))+ c45 ∑ Λ1⊕Ω1=n (191x2 2 − 4(Λ1⊕Υ1)) + c46 ∑ Λ1⊕Ω1=n (191x2 3 − 8(Λ1⊕Ω1))+ c47 ∑ Λ1⊕Ω1=n (191x2 4 − 6(Λ1⊕Ω1))). The coefficients are the same coefficients with preceding theorem, see Table 2 in [5]. Proof. It follows from the preceding theorem. References [1] M. Bhargava. On the Conway-Schneeberger fifteen theorem. Contemporary Mathematics 272, 27–37. 2000. REFERENCES 468 [2] M. Bhargava and J. Hanke. Universal quadratic forms and the 290-theorem, Inventiones Mathematicae, to appear. [3] J. Hanke. Some Recent Results about Ternary Quadratic Forms, CRM Proceedings and Lecture Notes, Number Theory, American Mathematical Society, 147-165. 2002. [4] B. Kendirli. Cusp Forms in S4(Γ0(79)) and the number of representations of positive in- tegers by some direct sum of binary quadratic forms with discriminant −79, Bulletin of Korean Mathematical Society, 49, 3, 529-572. 2012. [5] B. Köklüce. Theta series and coefficients. www.fatih.edu.tr/~bkoklu e/CT191. htm. [6] T. Lam. The algebraic theory of quadratic forms. Mathematics Lecture Note Series, W. A. Benjamin,Inc., Reading, Mass., 1973. [7] L. J. Mordell. A new Waring’s problem with squares of linear forms, The Quarterly Journal of Mathematics Oxford 1, 276–288. 1930. [8] H. Petersson. Modulfunktionen und quadratische Formen, Springer-Verlag, Berlin- Heidelberg-New York, 1982.