# Cusp Forms inS4(Γ0(47)) and the Number of Representations of Positive Integers by Some Direct Sum of Binary Quadratic Forms with Discriminant −47

Authors: Barış Kendirli
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

---

## Abstract

A basis ofS4(Γ0(47)) is given and the formulas for the number of representations
of positive integers by some direct sum of the quadratic forms x12+x1x2+12x22, 2x12±x1x2+6x22, 3x12±x1x2+4x22 are determined.


---

## Body

## 1. Introduction

This paper is the correction of the paper [1]. (1)
It is stated thatdimS4(Γ0(47),1)=5 at page 643 in [1]. But this dimension is 11 as stated at page 299 in [2]. Therefore, the coefficients of power series in (2.2), (2.3), (2.5), (2.6), (2.7), (2.8), (2.10), (2.12), (2.14), (2.15), (2.20), and (2.22) have to be calculated up to z11, and Theorem 2.4 and consequently Theorem 2.7 are false since 5 vectors cannot be a basis of 11-dimensional vector space.(2)
The notationsGk(Γ,χ) and (k,Γ,χ) are both used as if they are different like at line 11 at page 638.(3)
The definitions of(1.1)Γ1(N),Γ(N)and℘(τ;Q(X),Pv(X),h)=∑ni≡hi(modN)Pv(n1,n2,…,nk)zN(1/N)Q(n1,n2,…,nk)
have never been used in the paper.(4)
The class number ofℚ(-23) is 3; therefore, only F1, Φ1, and their combinations have been examined and a basis of S4(Γ0(23),1) could be obtained. The authors in [1] also examined only two quadratic forms
(1.2)F1=x12+x1x2+12x22,G1=2x12+x1x2+6x22
and their combinations. But, by simple calculations, it is possible to see that these quadratic forms are not enough to get a basis of S4(Γ0(47),1). The class number of ℚ(-47) is 5; therefore,
(1.3)F1=x12+x1x2+12x22,G1=2x12+x1x2+6x22,H1=2x12+x1x2+6x22
and their combinations have to be examined. Only in that case, it is possible to obtain a basis of S4(Γ0(47),1) as we have done in the following.


## 2. Determination of a Basis ofS4(Γ0(47))

We can calculate all reduced forms of a positive definite quadratic form(2.1)Q=ax2+bxy+cy2,a>0,
with discriminant Δ=-47 as follows:(2.2)F1=x12+x1x2+12x22,H1=2x12+x1x2+6x22,G1=3x12+x1x2+4x22,G1′=3x12-x1x2+4x22,H1′=2x12-x1x2+6x22.HereG1′ is the inverse of G1, and they represent the same integers. Similarly, H1′ is the inverse of H1 and they represent the same integers. Therefore, the theta series of H1 and H1′ are the same with the theta series of  G1 and  G1′, respectively. F1 is the identity element. It can be seen easily that, the group of these quadratic forms is a group of order 5 and can be described as(2.3)H12=G1′,H13=G1,H14=H1′,H15=F1.
We can easily see that for the quadratic forms(2.4)F1,G1,H1,
the determinant, the discriminant, and the character are(2.5)D=47,Δ=(-1)2/247=-47,χ(d)=(-47d).
Consequently, their theta series are in (2.6)M1(Γ0(47),(-47d)).
Hence by Theorem 2.1 in [3], F2, H2,  G2, F1⊕H1, F1⊕G1, and  H1⊕G1 are quadratic forms whose theta series are in (2.7)M2(Γ0(47)).We immediately obtain the following Corollary by Theorem 2.2 in [3].Corollary 2.1.
LetQ be a positive definite form of 8 variables whose theta series ΘQ is in
(2.8)M4(Γ0(47)).
Then the Eisenstein part of ΘQ is
(2.9)E(q:Q)=1+∑n=1∞(ασ3(n)qn+βσ3(n)q47n),
where
(2.10)ρ4=3!(2π)4ζ(4)=1240,α=240472-1474-1=24221,β=240474-472474-1=47224221,E(q:F4)=E(q:F3⊕H1)=E(q:F2⊕H2)=E(q:F1⊕H3)=E(q:H4)=E(q:F3⊕G1)=E(q:F2⊕G2)=E(q:F1⊕G3)=E(q:G4)=E(q:H3⊕G1)=E(q:H2⊕G2)=E(q:H1⊕G3)=1+24221∑n=1∞(qn+472q47n)σ3(n)=1+24221∑n=1∞σ3*(n)qn=1+24221q+24⋅9221q2+24⋅28221q3+24⋅73221q4+24⋅126221q5+24⋅252221q6+24⋅344221q7+24⋅585221q8+24⋅757221q9+24⋅1134221q10+24⋅1332221q11+⋯.
Here
(2.11)σ3*(n)={σ3(n)ifn≥1and47∤n,σ3(n)+472σ3(n47)if47∣n.Now we will determine the sum of quadratic formsF1, H1, and  G1 and select 11 spherical functions such that the corresponding cusp forms are linearly independent.(1)
For quadratic form(2.12)2F2=2x12+2x1x2+24x22+2x32+2x3x4+24x42=(x1,x2,x3,x4)(2112421124)(x1x2x3x4),
the determinant and a cofactor are
(2.13)D=472,A11=24⋅47.
By putting 2k=4, Q=F2, and appropriate i,j in Theorem 2.1 in [3], we get
(2.14)φ11=x12-1424⋅474722F2=x12-1247F2,
which will be spherical functions of second order with respect to F2.(2)
Similarly, for(2.15)2H2=4x12+2x1x2+12x22+4x32+2x3x4+12x42,
the determinant and some cofactors are
(2.16)D=472,A11=12⋅47,A12=-47,A13=12⋅12.
By putting 2k=4, Q=H2, and appropriate i,j in Theorem 2.1 in [3], we get
(2.17)φ11=x12-1412⋅474722H2=x12-647H2,φ12=x1x2+14474722H2=x1x2+12⋅47H2,φ13=x1x3-1412⋅124722H2=x1x3-72472H2,
which will be spherical functions of second order with respect to  H2.(3)
Similarly, for quadratic form(2.18)2G2=6x12+2x1x2+8x22+6x32+2x3x4+8x42,
the determinant and some cofactors are
(2.19)D=472,A11=8⋅47,A22=6⋅47,A33=8⋅47,A34=-47.
By putting 2k=4, Q=G2, and appropriate i,j in Theorem 2.1 in [3], we get
(2.20)φ11=x12-148⋅474722G2=x12-447G2,φ22=x22-146⋅474722G2=x22-347G2,φ33=x32-148⋅474722G2=x32-447G2,φ34=x3x4+14474722G2=x3x4+12⋅47G2,
which will be spherical functions of second order with respect to  G2.(4)
Similarly, for quadratic form(2.21)2(H1⊕G1)=4x12+2x1x2+12x22+6x32+2x3x4+8x42,
the determinant and some cofactors are
(2.22)D=472,A11=12⋅47,A22=4⋅47,A33=8⋅47.
By putting 2k=4, Q=H1⊕G1, and appropriate i,j in Theorem 2.1 in [3], we get
(2.23)φ11=x12-1412⋅474722(H1⊕G1)=x12-647(H1⊕G1),φ22=x22-144⋅474722(H1⊕G1)=x22-247(H1⊕G1),φ33=x22-148⋅474722(H1⊕G1)=x22-447(H1⊕G1),
which will be spherical functions of second order with respect to  H1⊕G1.Now we can determine a basis ofS4(Γ0(47)) whose dimension is 11, see [2].Theorem 2.2.
The following generalized 11 theta series:(2.24)ΘF2,φ11(q)=147∑n=1∞∑F2=n(47x12-12F2)qn,ΘH2,φ11(q)=147∑n=1∞∑H2=n(47x12-6H2)qn,ΘH2,φ12(q)=12⋅47∑n=1∞∑H2=n(2⋅47x1x2+H2)qn,ΘH2,φ13(q)=1472∑n=1∞∑H2=n(472x1x3-72H2)qn,ΘG2,φ11(q)=147∑n=1∞∑G2=n(47x12-4G2)qn,ΘG2,φ22(q)=147∑n=1∞∑G2=n(47x22-3G2)qn,ΘG2,φ33(q)=147∑n=1∞∑G2=n(47x32-4G2)qn,ΘG2,φ34(q)=12⋅47∑n=1∞∑G2=n(2⋅47x3x4+G2)qn,ΘH1⊕G1,φ11(q)=147∑n=1∞∑H1⊕G1=n(47x12-6(H1⊕G1))qn,ΘH1⊕G1,φ22(q)=147∑n=1∞∑H1⊕G1=n(47x22-2(H1⊕G1)),ΘH1⊕G1,φ33(q)=147∑n=1∞∑H1⊕G1=n(47x32-4(H1⊕G1))
are a basis of S4(Γ0(47)).Proof.
The series are cusp forms because of Theorem 2.1 in [3]. Moreover, by simple calculations, we have
(2.25)ΘF2,φ11(q)=179(46q+192q2+184q4+460q5+368q8+414q9+920q10+0q11+⋯),ΘH2,φ11(q)=147(46q2+92q4-144q6-74q7-12q8-178q9+460q10-152q11+⋯),ΘH2,φ12(q)=12⋅47(8q2+16q4+24q6-160q7+96q8-80q9+80q10+464q11+⋯),ΘH2,φ13(q)=1472(8q2+16q4+24q6+28q7+96q8+108q9+80q10+88q11+⋯),ΘG2,φ11(q)=147(46q3-64q4-4q6-36q7-162q8+88q9-132q10+24q11+⋯),ΘG2,φ22(q)=147(-48q3+30q4-98q6-36q7+26q8-100q9+56q10-164q11+⋯),ΘG2,φ33(q)=147(46q3-64q4+90q6-36q7-162q8+840q9-132q10+24q11+⋯),ΘG2,φ34(q)=147(-48q3-64q4-286q6-36q7-162q8-476q9-508q10-164q11+⋯),ΘH1⊕G1,φ11(q)=147(70q2-36q3-48q4+68q5-100q6+10q7+180q8-230q9-344q10-152q11+⋯),ΘH1⊕G1,φ22(q)=147(-24q2-36q3-48q4-120q5-194q6+10q7-384q8-42q9-344q10-340q11+⋯),ΘH1⊕G1,φ33(q)=147(-16q2+70q3-32q4+108q5-98q6-56q7+26q8-28q9-104q10-164q11+⋯).
The determinant of the coefficients matrix is 5321028802318956232704/4711. So, the set of theta series in the Theorem is a basis of S4(Γ0(47)).


## 3. Representation Numbers ofn

Proposition 3.1.
The theta series of the quadratic forms are(3.1)ΘF4(q)=ΘF2(q)ΘF2(q)=1+8q+24q2+32q3+24q4+48q5+96q6+64q7+24q8+104q9+144q10+96q11+⋯,ΘH4(q)=ΘH2(q)ΘH2(q)=1+8q2+24q4+40q6+8q7+72q8+56q9+144q10+144q11+⋯,ΘG4(q)=ΘG2(q)ΘG2(q)=1+8q3+8q4+32q6+48q7+32q8+80q9+144q10+144q11+⋯,ΘF3⊕H1(q)=ΘF3(q)ΘH1(q)=1+6q+14q2+20q3+30q4+40q5+38q6+62q7+98q8+84q9+112q10+184q11+⋯,ΘF2⊕H2(q)=ΘF2(q)ΘH2(q)=1+4q+8q2+16q3+24q4+24q5+36q6+52q7+64q8+112q9+144q10+152q11+⋯,ΘF1⊕H3(q)=ΘF1(q)ΘH3(q)=1+2q+6q2+12q3+14q4+24q5+26q6+34q7+66q8+92q9+136q10+168q11+⋯,ΘF3⊕G1(q)=ΘF3(q)ΘG1(q)=1+6q+12q2+10q3+20q4+60q5+66q6+40q7+98q8+154q9+108q10+112q11+⋯,ΘF2⊕G2(q)=ΘF2(q)ΘG2(q)=1+4q+4q2+4q3+24q4+40q5+24q6+56q7+124q8+108q9+112q10+184q11+⋯,ΘF1⊕G3(q)=ΘF1(q)ΘG3(q)=1+2q+6q3+20q4+12q5+18q6+72q7+78q8+70q9+148q10+192q11+⋯,ΘH3⊕G1(q)=ΘH3(q)ΘG1(q)=1+6q2+2q3+14q4+12q5+28q6+30q7+68q8+58q9+124q10+120q11+⋯,ΘH2⊕G2(q)=ΘH2(q)ΘG2(q)=1+4q2+4q3+8q4+16q5+28q6+28q7+68q8+68q9+112q10+144q11+⋯,ΘH1⊕G3(q)=ΘH1(q)ΘG3(q)=1+2q2+6q3+6q4+12q5+32q6+26q7+56q8+94q9+108q10+136q11+⋯,ΘF2⊕H1⊕G1(q)=ΘF2(q)⋅ΘH1⊕G1(q)=1+4q+6q2+10q3+22q4+28q5+40q6+74q7+76q8+82q9+148q10+168q11+⋯,ΘF1⊕H2⊕G1(q)=ΘH2(q)⋅ΘF1⊕G1(q)=1+2q+4q2+10q3+12q4+20q5+38q6+44q7+66q8+98q9+108q10+152q11+⋯,ΘF1⊕H1⊕G2(q)=ΘG2(q)⋅ΘF1⊕H1(q)=1+2q+2q2+8q3+14q4+16q5+38q6+54q7+54q8+104q9+144q10+144q11+⋯,
and the substraction of the any one of these theta series by the Eisenstein series
(3.2)E(q:F4)=⋯=E(q:F1⊕H1⊕G2)=1+24221∑n=1∞σ3*(n)qn=1+24221q+24⋅9221q2+24⋅28221q3+24⋅73221q4+24⋅126221q5+24⋅252221q6+24⋅344221q7+24⋅585221q8+24⋅757221q9+24⋅1134221q10+24⋅1332221q11+⋯
is a linear combinations of the theta series in the preceding theorem. The coefficients are given in table [4].Proof.
By determination of solutions of(3.3)F1=n,H1=n,G1=nforn=1,2,…,11,
we easily calculate the theta series
(3.4)ΘF1,ΘH1,ΘG1,ΘF2,ΘF3,ΘF4,ΘH2,ΘH3,ΘH4,ΘG2,ΘG3,ΘG4,ΘF1⊕H1,ΘF3⊕H1,ΘF2⊕H2,ΘF1⊕H3,ΘF1⊕G1,ΘF3⊕G1,ΘF4,ΘF2⊕G2,ΘF1⊕G3,ΘH1⊕G1,ΘH3⊕G1,ΘH2⊕G2,ΘH1⊕G3,ΘF2⊕H1⊕G1,ΘF1⊕H2⊕G1,ΘF1⊕H1⊕G2.
For the second part, now let us look at the case:
(3.5)ΘF4(q)-E(q:F)=1744221q+5088221q2+6400221q3+3552221q4+7584221q5+15168221q6+5888221q7-672221q8+4816221q9+4608221q10-10752221q11+⋯=c1ΘF2,φ11(q)+c2ΘH2,φ11(q)+c3ΘH2,φ12(q)+c4ΘH2,φ13(q)+c5ΘG2,φ11(q)+c6ΘG2,φ22(q)+c7ΘG2,φ33(q)+c8ΘG2,φ34(q)+c9ΘH1⊕G1,φ11(q)+c10ΘH1⊕G1,φ22(q)+c11ΘH1⊕G1,φ33(q).
By equating the coefficients of qn in both sides for n=1,2,3,…,11, we get an equation in coefficients
(3.6)(c1c2c3c4c5c6),(c7c8c9c10c11).
We repeat the same procedure for the other cases. At the end, by solving 11 linear equations in 11 variables, we get the coefficients in table [4].Corollary 3.2.
The representation numbers for the quadratic forms(3.7)Ω=F4,H4,H4′,G4,G4′,F3⊕H1,F3⊕H1′,F2⊕H2,F2⊕H2′,F1⊕H3,F1⊕H3′,F3⊕G1,F3⊕G1′,F2⊕G2,F2⊕G2′,F1⊕G3,F1⊕G3′,H3⊕G1,H3′⊕G1,H3⊕G1′,H3′⊕G1′,H2⊕G2,H2′⊕G2,H2⊕G2′,H2′⊕G2′,H1⊕G3,H1′⊕G3,H1⊕G3′,H1′⊕G3′,F2⊕H1⊕G1,F1⊕H2⊕G1,F1⊕H1⊕G2
are
(3.8)r(n;H)=24221σ3*(n)+c147∑F2=n(47x12-12n)+c247∑H2=n(47x12-6n)+c32⋅47∑H2=n(2⋅47x1x2+n)+c4472∑H2=n(472x1x3-72n)+c547∑G2=n(47x12-4n)+c647∑G2=n(47x22-3n)+c747∑G2=n(47x32-4n)+c82⋅47∑G2=n(2⋅47x3x4+n)+c947∑H1⊕G1=n(47x12-6n)+c1047∑H1⊕G1=n(47x22-2n)+c1147∑H1⊕G1=n(47x32-4n).
The coefficients
(3.9)(c1c2c3c4c5c6),(c7c8c9c10c11)
corresponding to the quadratic form Ω are given in [4].Proof.
It follows from the preceding  theorem.



---

*Source: 303492-2012-03-01.xml*