Cusp Forms in 𝑆_𝟒(Γ_𝟎(𝟒𝟕)) and the Number of Representations of Positive Integers by Some Direct Sum of Binary Quadratic Forms with Discriminant −𝟒𝟕

Abstract

A basis of 𝑆₄(Γ₀(47)) is given and the formulas for the number of representations of positive integers by some direct sum of the quadratic forms 𝑥²₁+𝑥₁𝑥₂+12𝑥²₂, 2𝑥²₁±𝑥₁𝑥₂+6𝑥²₂, 3𝑥²₁±𝑥₁𝑥₂+4𝑥²₂ are determined.

1. Introduction

This paper is the correction of the paper [1]. (1) It is stated that dim𝑆₄(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 notations 𝐺_𝑘(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 Γ₁(𝑁),Γ(𝑁)and℘𝜏;𝑄(𝑋),𝑃_𝑣(𝑋),ℎ=_{𝑛𝑖≡ℎ𝑖(mod𝑁)}𝑃_𝑣𝑛₁,𝑛₂,…,𝑛_𝑘𝑧^{(1/𝑁)𝑄(𝑛1,𝑛2,…,𝑛𝑘)}_𝑁 (1.1) have never been used in the paper.(4) The class number of ℚ(√−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 𝐹₁=𝑥²₁+𝑥₁𝑥₂+12𝑥²₂,𝐺₁=2𝑥²₁+𝑥₁𝑥₂+6𝑥²₂ (1.2) 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, 𝐹₁=𝑥²₁+𝑥₁𝑥₂+12𝑥²₂,𝐺₁=2𝑥²₁+𝑥₁𝑥₂+6𝑥²₂,𝐻₁=2𝑥²₁+𝑥₁𝑥₂+6𝑥²₂ (1.3) 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 of 𝑆₄(Γ₀(47))

We can calculate all reduced forms of a positive definite quadratic form𝑄=𝑎𝑥²+𝑏𝑥𝑦+𝑐𝑦²,𝑎>0, (2.1) with discriminant Δ=−47 as follows:𝐹₁=𝑥²₁+𝑥₁𝑥₂+12𝑥²₂,𝐻₁=2𝑥²₁+𝑥₁𝑥₂+6𝑥²₂,𝐺₁=3𝑥²₁+𝑥₁𝑥₂+4𝑥²₂,𝐺^′₁=3𝑥²₁−𝑥₁𝑥₂+4𝑥²₂,𝐻^′₁=2𝑥²₁−𝑥₁𝑥₂+6𝑥²₂. (2.2)

Here 𝐺^′₁ is the inverse of 𝐺₁, and they represent the same integers. Similarly, 𝐻^′₁ is the inverse of 𝐻₁ and they represent the same integers. Therefore, the theta series of 𝐻₁ and 𝐻^′₁ are the same with the theta series of 𝐺₁ and 𝐺^′₁, 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 as𝐻²₁=𝐺^′₁,𝐻³₁=𝐺₁,𝐻⁴₁=𝐻^′₁,𝐻⁵₁=𝐹₁. (2.3) We can easily see that for the quadratic forms𝐹₁,𝐺₁,𝐻₁, (2.4) the determinant, the discriminant, and the character are𝐷=47,Δ=(−1)^2/247=−47,𝜒(𝑑)=−47𝑑. (2.5) Consequently, their theta series are in 𝑀₁Γ₀(47),−47𝑑. (2.6) 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 𝑀₂Γ₀(47). (2.7)

We immediately obtain the following Corollary by Theorem 2.2 in [3].

Corollary 2.1. Let 𝑄 be a positive definite form of 8 variables whose theta series Θ_𝑄 is in 𝑀₄Γ₀(47). (2.8) Then the Eisenstein part of Θ_𝑄 is 𝐸𝑞∶𝑄=1+^∞_𝑛=1𝛼𝜎₃(𝑛)𝑞^𝑛+𝛽𝜎₃(𝑛)𝑞^47𝑛, (2.9) where 𝜌₄=3!(2𝜋)⁴𝜁(4)=1240,𝛼=24047²−147⁴=−124221,𝛽=24047⁴−47²47⁴−1=47²24,𝐸221𝑞∶𝐹₄=𝐸𝑞∶𝐹₃⊕𝐻₁=𝐸𝑞∶𝐹₂⊕𝐻₂=𝐸𝑞∶𝐹₁⊕𝐻₃=𝐸𝑞∶𝐻₄=𝐸𝑞∶𝐹₃⊕𝐺₁=𝐸𝑞∶𝐹₂⊕𝐺₂=𝐸𝑞∶𝐹₁⊕𝐺₃=𝐸𝑞∶𝐺₄=𝐸𝑞∶𝐻₃⊕𝐺₁=𝐸𝑞∶𝐻₂⊕𝐺₂=𝐸𝑞∶𝐻₁⊕𝐺₃=1+24221^∞_𝑛=1𝑞^𝑛+47²𝑞^47𝑛𝜎₃(𝑛)=1+24221^∞_𝑛=1𝜎^∗₃(𝑛)𝑞^𝑛=1+24221𝑞+24⋅9𝑞221²+24⋅28𝑞221³+24⋅73𝑞221⁴+24⋅126𝑞221⁵+24⋅252𝑞221⁶+24⋅344𝑞221⁷+24⋅585𝑞221⁸+24⋅757𝑞221⁹+24⋅1134𝑞221¹⁰+24⋅1332𝑞221¹¹+⋯. (2.10) Here 𝜎^∗₃(𝑛)=𝜎₃(𝑛)𝜎𝑖𝑓𝑛≥1𝑎𝑛𝑑47∤𝑛,₃(𝑛)+47²𝜎₃𝑛47𝑖𝑓47∣𝑛. (2.11)

Now we will determine the sum of quadratic forms 𝐹₁, 𝐻₁, and 𝐺₁ and select 11 spherical functions such that the corresponding cusp forms are linearly independent.(1) For quadratic form 2𝐹₂=2𝑥²₁+2𝑥₁𝑥₂+24𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+24𝑥²₄=𝑥₁,𝑥₂,𝑥₃,𝑥₄21𝑥12421124¹𝑥₂𝑥₃𝑥₄, (2.12) the determinant and a cofactor are 𝐷=47²,𝐴₁₁=24⋅47. (2.13) By putting 2𝑘=4, 𝑄=𝐹₂, and appropriate 𝑖,𝑗 in Theorem 2.1 in [3], we get 𝜑₁₁=𝑥²₁−1424⋅4747²2𝐹₂=𝑥²₁−12𝐹47², (2.14) which will be spherical functions of second order with respect to 𝐹₂.(2)Similarly, for 2𝐻₂=4𝑥²₁+2𝑥₁𝑥₂+12𝑥²₂+4𝑥²₃+2𝑥₃𝑥₄+12𝑥²₄, (2.15) the determinant and some cofactors are 𝐷=47²,𝐴₁₁=12⋅47,𝐴₁₂=−47,𝐴₁₃=12⋅12. (2.16) By putting 2𝑘=4, 𝑄=𝐻₂, and appropriate 𝑖,𝑗 in Theorem 2.1 in [3], we get 𝜑₁₁=𝑥²₁−1412⋅4747²2𝐻₂=𝑥²₁−6𝐻47²,𝜑₁₂=𝑥₁𝑥₂+144747²2𝐻₂=𝑥₁𝑥₂+1𝐻2⋅47²,𝜑₁₃=𝑥₁𝑥₃−1412⋅1247²2𝐻₂=𝑥₁𝑥₃−7247²𝐻₂, (2.17) which will be spherical functions of second order with respect to 𝐻₂.(3) Similarly, for quadratic form 2𝐺₂=6𝑥²₁+2𝑥₁𝑥₂+8𝑥²₂+6𝑥²₃+2𝑥₃𝑥₄+8𝑥²₄, (2.18) the determinant and some cofactors are 𝐷=47²,𝐴₁₁=8⋅47,𝐴₂₂=6⋅47,𝐴₃₃=8⋅47,𝐴₃₄=−47. (2.19) By putting 2𝑘=4, 𝑄=𝐺₂, and appropriate 𝑖,𝑗 in Theorem 2.1 in [3], we get 𝜑₁₁=𝑥²₁−148⋅4747²2𝐺₂=𝑥²₁−4𝐺47²,𝜑₂₂=𝑥²₂−146⋅4747²2𝐺₂=𝑥²₂−3𝐺47²,𝜑₃₃=𝑥²₃−148⋅4747²2𝐺₂=𝑥²₃−4𝐺47²,𝜑₃₄=𝑥₃𝑥₄+144747²2𝐺₂=𝑥₃𝑥₄+1𝐺2⋅47², (2.20) which will be spherical functions of second order with respect to 𝐺₂.(4) Similarly, for quadratic form 2𝐻₁⊕𝐺₁=4𝑥²₁+2𝑥₁𝑥₂+12𝑥²₂+6𝑥²₃+2𝑥₃𝑥₄+8𝑥²₄, (2.21) the determinant and some cofactors are 𝐷=47²,𝐴₁₁=12⋅47,𝐴₂₂=4⋅47,𝐴₃₃=8⋅47. (2.22) By putting 2𝑘=4, 𝑄=𝐻₁⊕𝐺₁, and appropriate 𝑖,𝑗 in Theorem 2.1 in [3], we get 𝜑₁₁=𝑥²₁−1412⋅4747²2𝐻₁⊕𝐺₁=𝑥²₁−6𝐻47¹⊕𝐺₁,𝜑₂₂=𝑥²₂−144⋅4747²2𝐻₁⊕𝐺₁=𝑥²₂−2𝐻47¹⊕𝐺₁,𝜑₃₃=𝑥²₂−148⋅4747²2𝐻₁⊕𝐺₁=𝑥²₂−4𝐻47¹⊕𝐺₁, (2.23) which will be spherical functions of second order with respect to 𝐻₁⊕𝐺₁.

Now we can determine a basis of 𝑆₄(Γ₀(47)) whose dimension is 11, see [2].

Theorem 2.2. The following generalized 11 theta series: Θ_𝐹2,𝜑11𝑞=147^∞_{𝑛=1𝐹2=𝑛}47𝑥²₁−12𝐹₂𝑞^𝑛,Θ_𝐻2,𝜑11𝑞=147^∞_{𝑛=1𝐻2=𝑛}47𝑥²₁−6𝐻₂𝑞^𝑛,Θ_𝐻2,𝜑12𝑞=12⋅47^∞_{𝑛=1𝐻2=𝑛}2⋅47𝑥₁𝑥₂+𝐻₂𝑞^𝑛,Θ_𝐻2,𝜑13𝑞=147^2∞_{𝑛=1𝐻2=𝑛}47²𝑥₁𝑥₃−72𝐻₂𝑞^𝑛,Θ_𝐺2,𝜑11𝑞=147^∞_{𝑛=1𝐺2=𝑛}47𝑥²₁−4𝐺₂𝑞^𝑛,Θ_𝐺2,𝜑22𝑞=147^∞_{𝑛=1𝐺2=𝑛}47𝑥²₂−3𝐺₂𝑞^𝑛,Θ_𝐺2,𝜑33𝑞=147^∞_{𝑛=1𝐺2=𝑛}47𝑥²₃−4𝐺₂𝑞^𝑛,Θ_𝐺2,𝜑34𝑞=12⋅47^∞_{𝑛=1𝐺2=𝑛}2⋅47𝑥₃𝑥₄+𝐺₂𝑞^𝑛,Θ_{𝐻1⊕𝐺1,𝜑11}𝑞=147^∞_{𝑛=1𝐻1⊕𝐺1=𝑛}47𝑥²₁−6𝐻₁⊕𝐺₁𝑞^𝑛,Θ_{𝐻1⊕𝐺1,𝜑22}𝑞=147^∞_{𝑛=1𝐻1⊕𝐺1=𝑛}47𝑥²₂−2𝐻₁⊕𝐺₁,Θ_{𝐻1⊕𝐺1,𝜑33}𝑞=147^∞_{𝑛=1𝐻1⊕𝐺1=𝑛}47𝑥²₃−4𝐻₁⊕𝐺₁ (2.24) 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 Θ_𝐹2,𝜑11𝑞=17946𝑞+192𝑞²+184𝑞⁴+460𝑞⁵+368𝑞⁸+414𝑞⁹+920𝑞¹⁰+0𝑞¹¹,Θ+⋯_𝐻2,𝜑11𝑞=14746𝑞²+92𝑞⁴−144𝑞⁶−74𝑞⁷−12𝑞⁸−178𝑞⁹+460𝑞¹⁰−152𝑞¹¹,Θ+⋯_𝐻2,𝜑12𝑞=12⋅478𝑞²+16𝑞⁴+24𝑞⁶−160𝑞⁷+96𝑞⁸−80𝑞⁹+80𝑞¹⁰+464𝑞¹¹,Θ+⋯_𝐻2,𝜑13𝑞=147²8𝑞²+16𝑞⁴+24𝑞⁶+28𝑞⁷+96𝑞⁸+108𝑞⁹+80𝑞¹⁰+88𝑞¹¹,Θ+⋯_𝐺2,𝜑11𝑞=14746𝑞³−64𝑞⁴−4𝑞⁶−36𝑞⁷−162𝑞⁸+88𝑞⁹−132𝑞¹⁰+24𝑞¹¹,Θ+⋯_𝐺2,𝜑22𝑞=147−48𝑞³+30𝑞⁴−98𝑞⁶−36𝑞⁷+26𝑞⁸−100𝑞⁹+56𝑞¹⁰−164𝑞¹¹,Θ+⋯_𝐺2,𝜑33𝑞=14746𝑞³−64𝑞⁴+90𝑞⁶−36𝑞⁷−162𝑞⁸+840𝑞⁹−132𝑞¹⁰+24𝑞¹¹,Θ+⋯_𝐺2,𝜑34𝑞=147−48𝑞³−64𝑞⁴−286𝑞⁶−36𝑞⁷−162𝑞⁸−476𝑞⁹−508𝑞¹⁰−164𝑞¹¹,Θ+⋯_{𝐻1⊕𝐺1,𝜑11}𝑞=14770𝑞²−36𝑞³−48𝑞⁴+68𝑞⁵−100𝑞⁶+10𝑞⁷+180𝑞⁸−230𝑞⁹−344𝑞¹⁰−152𝑞¹¹,Θ+⋯_{𝐻1⊕𝐺1,𝜑22}𝑞=147−24𝑞²−36𝑞³−48𝑞⁴−120𝑞⁵−194𝑞⁶+10𝑞⁷−384𝑞⁸−42𝑞⁹−344𝑞¹⁰−340𝑞¹¹,Θ+⋯_{𝐻1⊕𝐺1,𝜑33}𝑞=147−16𝑞²+70𝑞³−32𝑞⁴+108𝑞⁵−98𝑞⁶−56𝑞⁷+26𝑞⁸−28𝑞⁹−104𝑞¹⁰−164𝑞¹¹+⋯. (2.25) 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 of 𝑛

Proposition 3.1. The theta series of the quadratic forms are Θ_𝐹4𝑞=Θ_𝐹2𝑞Θ_𝐹2𝑞=1+8𝑞+24𝑞²+32𝑞³+24𝑞⁴+48𝑞⁵+96𝑞⁶+64𝑞⁷+24𝑞⁸+104𝑞⁹+144𝑞¹⁰+96𝑞¹¹Θ+⋯,_𝐻4𝑞=Θ_𝐻2𝑞Θ_𝐻2𝑞=1+8𝑞²+24𝑞⁴+40𝑞⁶+8𝑞⁷+72𝑞⁸+56𝑞⁹+144𝑞¹⁰+144𝑞¹¹Θ+⋯,_𝐺4𝑞=Θ_𝐺2𝑞Θ_𝐺2𝑞=1+8𝑞³+8𝑞⁴+32𝑞⁶+48𝑞⁷+32𝑞⁸+80𝑞⁹+144𝑞¹⁰+144𝑞¹¹Θ+⋯,_{𝐹3⊕𝐻1}𝑞=Θ_𝐹3𝑞Θ_𝐻1𝑞=1+6𝑞+14𝑞²+20𝑞³+30𝑞⁴+40𝑞⁵+38𝑞⁶+62𝑞⁷+98𝑞⁸+84𝑞⁹+112𝑞¹⁰+184𝑞¹¹Θ+⋯,_{𝐹2⊕𝐻2}𝑞=Θ_𝐹2𝑞Θ_𝐻2𝑞=1+4𝑞+8𝑞²+16𝑞³+24𝑞⁴+24𝑞⁵+36𝑞⁶+52𝑞⁷+64𝑞⁸+112𝑞⁹+144𝑞¹⁰+152𝑞¹¹Θ+⋯,_{𝐹1⊕𝐻3}𝑞=Θ_𝐹1𝑞Θ_𝐻3𝑞=1+2𝑞+6𝑞²+12𝑞³+14𝑞⁴+24𝑞⁵+26𝑞⁶+34𝑞⁷+66𝑞⁸+92𝑞⁹+136𝑞¹⁰+168𝑞¹¹Θ+⋯,_{𝐹3⊕𝐺1}𝑞=Θ_𝐹3𝑞Θ_𝐺1𝑞=1+6𝑞+12𝑞²+10𝑞³+20𝑞⁴+60𝑞⁵+66𝑞⁶+40𝑞⁷+98𝑞⁸+154𝑞⁹+108𝑞¹⁰+112𝑞¹¹Θ+⋯,_{𝐹2⊕𝐺2}𝑞=Θ_𝐹2𝑞Θ_𝐺2𝑞=1+4𝑞+4𝑞²+4𝑞³+24𝑞⁴+40𝑞⁵+24𝑞⁶+56𝑞⁷+124𝑞⁸+108𝑞⁹+112𝑞¹⁰+184𝑞¹¹Θ+⋯,_{𝐹1⊕𝐺3}𝑞=Θ_𝐹1𝑞Θ_𝐺3𝑞=1+2𝑞+6𝑞³+20𝑞⁴+12𝑞⁵+18𝑞⁶+72𝑞⁷+78𝑞⁸+70𝑞⁹+148𝑞¹⁰+192𝑞¹¹Θ+⋯,_{𝐻3⊕𝐺1}𝑞=Θ_𝐻3𝑞Θ_𝐺1𝑞=1+6𝑞²+2𝑞³+14𝑞⁴+12𝑞⁵+28𝑞⁶+30𝑞⁷+68𝑞⁸+58𝑞⁹+124𝑞¹⁰+120𝑞¹¹Θ+⋯,_{𝐻2⊕𝐺2}𝑞=Θ_𝐻2𝑞Θ_𝐺2𝑞=1+4𝑞²+4𝑞³+8𝑞⁴+16𝑞⁵+28𝑞⁶+28𝑞⁷+68𝑞⁸+68𝑞⁹+112𝑞¹⁰+144𝑞¹¹Θ+⋯,_{𝐻1⊕𝐺3}𝑞=Θ_𝐻1𝑞Θ_𝐺3𝑞=1+2𝑞²+6𝑞³+6𝑞⁴+12𝑞⁵+32𝑞⁶+26𝑞⁷+56𝑞⁸+94𝑞⁹+108𝑞¹⁰+136𝑞¹¹Θ+⋯,_{𝐹2⊕𝐻1⊕𝐺1}𝑞=Θ_𝐹2𝑞⋅Θ_{𝐻1⊕𝐺1}𝑞=1+4𝑞+6𝑞²+10𝑞³+22𝑞⁴+28𝑞⁵+40𝑞⁶+74𝑞⁷+76𝑞⁸+82𝑞⁹+148𝑞¹⁰+168𝑞¹¹Θ+⋯,_{𝐹1⊕𝐻2⊕𝐺1}𝑞=Θ_𝐻2𝑞⋅Θ_{𝐹1⊕𝐺1}𝑞=1+2𝑞+4𝑞²+10𝑞³+12𝑞⁴+20𝑞⁵+38𝑞⁶+44𝑞⁷+66𝑞⁸+98𝑞⁹+108𝑞¹⁰+152𝑞¹¹Θ+⋯,_{𝐹1⊕𝐻1⊕𝐺2}𝑞=Θ_𝐺2𝑞⋅Θ_{𝐹1⊕𝐻1}𝑞=1+2𝑞+2𝑞²+8𝑞³+14𝑞⁴+16𝑞⁵+38𝑞⁶+54𝑞⁷+54𝑞⁸+104𝑞⁹+144𝑞¹⁰+144𝑞¹¹+⋯, (3.1) and the substraction of the any one of these theta series by the Eisenstein series 𝐸𝑞∶𝐹₄=⋯=𝐸𝑞∶𝐹₁⊕𝐻₁⊕𝐺₂=1+24221^∞_𝑛=1𝜎^∗₃(𝑛)𝑞^𝑛=1+24221𝑞+24⋅9𝑞221²+24⋅28𝑞221³+24⋅73𝑞221⁴+24⋅126𝑞221⁵+24⋅252𝑞221⁶+24⋅344𝑞221⁷+24⋅585𝑞221⁸+24⋅757𝑞221⁹+24⋅1134𝑞221¹⁰+24⋅1332𝑞221¹¹+⋯ (3.2) 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 𝐹₁=𝑛,𝐻₁=𝑛,𝐺₁=𝑛for𝑛=1,2,…,11, (3.3) we easily calculate the theta series Θ_𝐹1,Θ_𝐻1,Θ_𝐺1,Θ_𝐹2,Θ_𝐹3,Θ_𝐹4,Θ_𝐻2,Θ_𝐻3,Θ_𝐻4,Θ_𝐺2,Θ_𝐺3,Θ_𝐺4,Θ_{𝐹1⊕𝐻1},Θ_𝐹3⊕H1,Θ_{𝐹2⊕𝐻2},Θ_{𝐹1⊕𝐻3},Θ_{𝐹1⊕𝐺1},Θ_{𝐹3⊕𝐺1},Θ_𝐹4,Θ_{𝐹2⊕𝐺2},Θ_{𝐹1⊕𝐺3},Θ_{𝐻1⊕𝐺1},Θ_{𝐻3⊕𝐺1},Θ_{𝐻2⊕𝐺2},Θ_{𝐻1⊕𝐺3},Θ_{𝐹2⊕𝐻1⊕𝐺1},Θ_{𝐹1⊕𝐻2⊕𝐺1},Θ_{𝐹1⊕𝐻1⊕𝐺2}. (3.4) For the second part, now let us look at the case: Θ_𝐹4𝑞−𝐸𝑞∶𝐹=1744221𝑞+5088𝑞221²+6400𝑞221³+3552𝑞221⁴+7584𝑞221⁵+15168𝑞221⁶+5888𝑞221⁷−672𝑞221⁸+4816𝑞221⁹+4608𝑞221¹⁰−10752𝑞221¹¹+⋯=𝑐₁Θ_𝐹2,𝜑11𝑞+𝑐₂Θ_𝐻2,𝜑11𝑞+𝑐₃Θ_𝐻2,𝜑12𝑞+𝑐₄Θ_𝐻2,𝜑13𝑞+𝑐₅Θ_𝐺2,𝜑11𝑞+𝑐₆Θ_𝐺2,𝜑22𝑞+𝑐₇Θ_𝐺2,𝜑33𝑞+𝑐₈Θ_𝐺2,𝜑34𝑞+𝑐₉Θ_{𝐻1⊕𝐺1,𝜑11}𝑞+𝑐₁₀Θ_{𝐻1⊕𝐺1,𝜑22}𝑞+𝑐₁₁Θ_{𝐻1⊕𝐺1,𝜑33}𝑞. (3.5) By equating the coefficients of 𝑞^𝑛 in both sides for 𝑛=1,2,3,…,11, we get an equation in coefficients 𝑐₁𝑐₂𝑐₃𝑐₄𝑐₅𝑐₆,𝑐₇𝑐₈𝑐₉𝑐₁₀𝑐₁₁. (3.6) 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 Ω=𝐹₄,𝐻₄,𝐻^′₄,𝐺₄,𝐺^′₄,𝐹₃⊕𝐻₁,𝐹₃⊕𝐻^′₁,𝐹₂⊕𝐻₂,𝐹₂⊕𝐻^′₂,𝐹₁⊕𝐻₃,𝐹₁⊕𝐻^′₃,𝐹₃⊕𝐺₁,𝐹₃⊕𝐺^′₁,𝐹₂⊕𝐺₂,𝐹₂⊕𝐺^′₂,𝐹₁⊕𝐺₃,𝐹₁⊕𝐺^′₃,𝐻₃⊕𝐺₁,𝐻^′₃⊕𝐺₁,𝐻₃⊕𝐺^′₁,𝐻^′₃⊕𝐺^′₁,𝐻₂⊕𝐺₂,𝐻^′₂⊕𝐺₂,𝐻₂⊕𝐺^′₂,𝐻^′₂⊕𝐺^′₂,𝐻₁⊕𝐺₃,𝐻^′₁⊕𝐺₃,𝐻₁⊕𝐺^′₃,𝐻^′₁⊕𝐺^′₃,𝐹₂⊕𝐻₁⊕𝐺₁,𝐹₁⊕𝐻₂⊕𝐺₁,𝐹₁⊕𝐻₁⊕𝐺₂ (3.7) are 𝑟(𝑛;𝐻)=24𝜎221^∗3(𝑛)+𝑐₁47_𝐹2=𝑛47𝑥²₁+𝑐−12𝑛²47_𝐻2=𝑛47𝑥²₁+𝑐−6𝑛₃2⋅47_𝐻2=𝑛2⋅47𝑥₁𝑥₂+𝑐+𝑛⁴47²_𝐻2=𝑛47²𝑥₁𝑥₃+𝑐−72𝑛⁵47_𝐺2=𝑛47𝑥²₁+𝑐−4𝑛₆47_𝐺2=𝑛47𝑥²₂+𝑐−3𝑛⁷47_𝐺2=𝑛47𝑥²₃+𝑐−4𝑛⁸2⋅47_𝐺2=𝑛2⋅47𝑥₃𝑥₄+𝑐+𝑛₉47_{𝐻1⊕𝐺1=𝑛}47𝑥²₁+𝑐−6𝑛¹⁰47_{𝐻1⊕𝐺1=𝑛}47𝑥²₂+𝑐−2𝑛¹¹47_{𝐻1⊕𝐺1=𝑛}47𝑥²₃−4𝑛. (3.8) The coefficients 𝑐₁𝑐₂𝑐₃𝑐₄𝑐₅𝑐₆,𝑐₇𝑐₈𝑐₉𝑐₁₀𝑐₁₁ (3.9) corresponding to the quadratic form Ω are given in [4].

Proof. It follows from the preceding theorem.