The Bases of 𝑀₄(Γ₀(71)), 𝑀₆(Γ₀(71)) and the Number of Representation of Integers

Abstract

Following a fundamental theorem of Hecke, some bases of 𝑆₄(Γ₀(71)) and 𝑆₆(Γ₀(71)) are determined, and explicit formulas are obtained for the number of representations of positive integers by all possible direct sums (111 different combinations) of seven quadratic forms from the class group of equivalence classes of quadratic forms with discriminant −71 whose representatives are 𝑥²₁+𝑥₁𝑥₂+18𝑥²₂,2𝑥²₁±𝑥₁𝑥₂+9𝑥²₂,3𝑥²₁±𝑥₁𝑥₂+6𝑥²₂,and4𝑥²₁±3𝑥₁𝑥₂+5𝑥²₂.

1. Introduction

In this work, we will obtain the representation number of certain positive definite quadratic forms by means of the deep theorems of Hecke [1] and Schoeneberg [2]. Lomadze [3] and Vepkhvadze [4] have studied this subject for several quadratic forms. Also the works of Kani [5], Sun and Williams [6] clarify and contribute the subject. Of couse, the work of Hijikata et al. [7] for the construction of a general bases of modular forms of Γ₀(𝑁) [7] is very important work. Here, we have found bases in special cases 𝑀₄(Γ₀(71)),𝑀₆(Γ₀(71)) by extending our work on [8, 9] to the case of discriminant −71 including weight 6 case.

There exist 7 equivalence classes of binary quadratic forms of discriminant −71 whose reduced forms are 𝐹₁=𝑥²₁+𝑥₁𝑥₂+18𝑥²₂,Φ₁=2𝑥²₁−𝑥₁𝑥₂+9𝑥²₂,Φ^′₁=2𝑥²₁+𝑥₁𝑥₂+9𝑥²₂,Ψ₁=3𝑥²₁−𝑥₁𝑥₂+6𝑥²₂,Ψ^′₁=3𝑥²₁+𝑥₁𝑥₂+6𝑥²₂,Λ₁=4𝑥²₁−3𝑥₁𝑥₂+5𝑥²₂,Λ^′₁=4𝑥²₁+3𝑥₁𝑥₂+5𝑥²₂. (1)

Here, Φ^′₁ is the inverse of Φ₁, and they represent the same integers. Similarly, Ψ^′₁ is the inverse of Ψ₁, and they represent the same integers, and Λ^′₁ is the inverse of Λ₁, and they represent the same integers. The group of these quadratic forms is a group of order 7 and can be described, see [10], easily as Φ₁,Φ²₁=Λ^′₁,Φ³₁=Ψ₁,Φ⁴₁=Ψ^′₁,Φ⁵₁=Λ₁,Φ⁶₁=Φ^′₁,Φ⁷₁=𝐹₁. (2) Therefore, the theta series of Φ₁, Φ^′₁ and Ψ₁, Ψ^′₁ and Λ₁ and Λ^′₁ are the same, respectively. 𝐹₁ is the identity element. Since 71 is prime, there is only one genus, that is, principal genus.

Let 𝐹_𝑘=𝐹₁+⋯+𝐹₁(𝑘times),Φ_𝑘=Φ₁+⋯+Φ₁(𝑘times),Ψ_𝑘=Ψ₁+⋯+Ψ₁(𝑘times),Λ_𝑘=Λ₁+⋯+Λ₁(𝑘times) (3) Be  𝑘 direct sums of the quadratic forms. In this paper, we will obtain the formulas of 𝑟(𝑛;𝑄) for the quadratic forms of 8 variables 𝐻=𝐹₄,Φ₄,Ψ₄,Λ₄,𝐹₃⊕Φ₁,𝐹₂⊕Φ₂,𝐹₁⊕Φ₃,𝐹₃⊕Ψ₁,𝐹₂⊕Ψ₂,𝐹₁⊕Ψ₃,Φ₃⊕Ψ₁,Φ₂⊕Ψ₂,Φ₁⊕Ψ₃,𝐹₃⊕Λ₁,𝐹₂⊕Λ₂,𝐹₁⊕Λ₃,Φ₃⊕Λ₁,Φ₂⊕Λ₂,Φ₁⊕Λ₃,Ψ₃⊕Λ₁,Ψ₂⊕Λ₂,Ψ₁⊕Λ₃,𝐹₂⊕Φ₁⊕Ψ₁,𝐹₂⊕Φ₁⊕Λ₁,𝐹₂⊕Ψ₁⊕Λ₁,𝐹₁⊕Φ₂⊕Ψ₁,𝐹₁⊕Φ₂⊕Λ₁,Ψ₁⊕Φ₂⊕Λ₁,𝐹₁⊕Λ₁⊕Ψ₂,𝐹₁⊕Φ₁⊕Ψ₂,Λ₁⊕Φ₁⊕Ψ₂,𝐹₁⊕Λ₁⊕Ψ₂,𝐹₁⊕Φ₁⊕Λ₂,𝐹₁⊕Ψ₁⊕Λ₂,Φ₁⊕Ψ₁⊕Λ₂,𝐹₁⊕Φ₁⊕Ψ₁⊕Λ₁, (4) and quadratic form of 12 variables 𝑄=𝐹₆,𝐹₅⊕Φ₁,𝐹₅⊕Ψ₁,𝐹₅⊕Λ₁,𝐹₄⊕Φ₂,𝐹₄⊕Ψ₂,𝐹₄⊕Λ₂,𝐹₄⊕Φ₁⊕Ψ₁,𝐹₄⊕Φ₁⊕Λ₁,𝐹₄⊕Ψ₁⊕Λ₁,𝐹₃⊕Φ₃,𝐹₃⊕Ψ₃,𝐹₃⊕Λ₃,𝐹₃⊕Φ₂⊕Ψ₁,𝐹₃⊕Φ₂⊕Λ₁,𝐹₃⊕Ψ₂⊕Φ₁,𝐹₃⊕Ψ₂⊕Λ₁,𝐹₃⊕Λ₂⊕Φ₁,𝐹₃⊕Λ₂⊕Ψ₁,𝐹₃⊕Ψ₂⊕Φ₁,𝐹₃⊕Φ₁⊕Ψ₁⊕Λ₁,𝐹₂⊕Φ₄,𝐹₂⊕Ψ₄,𝐹₂⊕Λ₄,𝐹₂⊕Φ₃⊕Ψ₁,𝐹₂⊕Φ₃⊕Λ₁,𝐹₂⊕Ψ₃⊕Φ₁,𝐹₂⊕Ψ₃⊕Λ₁,𝐹₂⊕Λ₃⊕Φ₁,𝐹₂⊕Λ₃⊕Ψ₁,𝐹₂⊕Φ₂⊕Ψ₁⊕Λ₁,𝐹₂⊕Ψ₂⊕Φ₁⊕Λ₁,𝐹₂⊕Λ₂⊕Φ₁⊕Ψ₁,𝐹₁⊕Φ₅,𝐹₁⊕Ψ₅,𝐹₁⊕Λ₅,𝐹₁⊕Φ₄⊕Ψ₁,𝐹₁⊕Φ₄⊕Λ₁,𝐹₁⊕Ψ₄⊕Φ₁,𝐹₁⊕Ψ₄⊕Λ₁,𝐹₁⊕Λ₄⊕Φ₁,𝐹₁⊕Λ₄⊕Ψ₁,𝐹₁⊕Φ₃⊕Ψ₂,𝐹₁⊕Φ₃⊕Λ₂,𝐹₁⊕Φ₃⊕Ψ₁⊕Λ₁,𝐹₁⊕Ψ₃⊕Φ₂,𝐹₁⊕Ψ₃⊕Λ₂,𝐹₁⊕Ψ₃⊕Φ₁⊕Λ₁,𝐹₁⊕Λ₃⊕Φ₂,𝐹₁⊕Λ₃⊕Ψ₂,𝐹₁⊕Λ₃⊕Φ₁⊕Ψ₁,𝐹₁⊕Φ₂⊕Ψ₂⊕Λ₁,𝐹₁⊕Φ₂⊕Λ₂⊕Ψ₁,Φ₆,Φ₅⊕Ψ₁,Φ₅⊕Λ₁,Φ₄⊕Ψ₂,Φ₄⊕Λ₂,Φ₄⊕Ψ₁⊕Λ₁,Φ₃⊕Ψ₃,Φ₃⊕Λ₃,Φ₃⊕Ψ₂⊕Λ₁,Φ₃⊕Λ₂⊕Ψ₁,Φ₂⊕Ψ₄,Φ₂⊕Λ₄,Φ₁⊕Ψ₅,Φ₁⊕Λ₅,Φ₁⊕Ψ₄⊕Λ₁,Φ₁⊕Λ₄⊕Ψ₁,Φ₁⊕Ψ₃⊕Λ₂,Φ₁⊕Λ₃⊕Ψ₂,Ψ₆,Ψ₅⊕Λ₁,Ψ₄⊕Λ₂,Ψ₃⊕Λ₃,Ψ₂⊕Λ₄,Ψ₁⊕Λ₅,Λ₆. (5)

In these formulas one can replace Φ₁,Λ₁, and Ψ₁ by Φ^′₁,Λ^′₁, and Ψ^′₁, respectively.

2. The Positive Definite Quadratic Forms

Now we will give some definitions, an important theorem and evaluation of our quadratic forms.

Definition 1. Let 𝑄∶ℤ^2𝑘→ℤ be a positive definite integer-valued form of 2𝑘 variables 𝑄=^2𝑘_{1≤𝑖≤𝑗≤2𝑘}𝑏_𝑖𝑗𝑥_𝑖𝑥_𝑗,𝑏_𝑖𝑗∈ℤ. (6) Let 𝐷 be the determinant of the quadratic form 2𝑄=^2𝑘_𝑖,𝑗=1𝑎_𝑖𝑗𝑥_𝑖𝑥_𝑗,𝑎_𝑖𝑖=2𝑏_𝑖𝑖,𝑎_𝑗𝑖=𝑎_𝑖𝑗=𝑏_𝑖𝑗for𝑖<𝑗; (7) And let 𝐴_𝑖𝑗 be the cofactors of 𝑎_𝑖𝑗 for 1≤𝑖,𝑗≤2𝑘. If 𝛿=gcd(𝐴_𝑖𝑖/2,𝐴_𝑖𝑗for1≤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. Then,
Δ=(−1)^𝑘𝐷 is called the discriminant of the form 𝑄.

Theorem 2. Let 𝑄∶ℤ^2𝑘→ℤ be a positive definite integer-valued form of 2𝑘 variables of level 𝑁 and discriminant Δ. Then (1) the theta function Θ_𝑄𝑞=_{(𝑛1,𝑛2,…,𝑛2𝑘)∈ℤ×ℤ⋯×ℤ}𝑞^{𝑄(𝑛1,𝑛2,…,𝑛2𝑘)}=1+∑^∞_𝑛=1𝑟(𝑛;𝑄)𝑞^𝑛,𝑞=𝑒^{2𝜋𝑖𝑧} (8) is a modular form on Γ₀(𝑁) of weight 𝑘 and Kronecker character 𝜒_Δ(𝑑)=Δ𝑑forℤ𝑑∈𝑁ℤ^𝑥, (9) that is, Θ_𝑄∈𝑀_𝑘(Γ₀(𝑁),𝜒_Δ). Consider the following:
(2) The homogeneous quadratic polynomials in 2𝑘 variables 𝜑_𝑖𝑗=𝑥_𝑖𝑥_𝑗−1𝐴2𝑘𝐷^𝑖𝑗2𝑄,1≤𝑖,𝑗≤2𝑘 (10) are spherical functions of second order with respect to 𝑄. Consider the following:
(3) The theta series Θ_{𝑄,𝜑𝑖𝑗}𝑞=^∞_{𝑛=1𝑄=𝑛}𝜑_𝑖𝑗𝑞^𝑛 (11) is a cusp form in 𝑆_𝑘+2(Γ₀(𝑁),𝜒_Δ).
(4) If two quadratic forms 𝑄₁,𝑄₂ have the same level 𝑁 and the characters 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 [1, 2].

Now, let us look at the positive definite quadratic forms of discriminant −71.

For the quadratic form 𝐹₁=𝑥²₁+𝑥₁𝑥₂+18𝑥²₂, 2𝐹₁=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂=𝑥₁,𝑥₂21𝑥136𝑥¹₂, (12) the determinant and a cofactor are 𝐷=71,𝐴₂₂=2. (13) So, 𝛿=1,𝑁=𝐷=71, and the discriminant is Δ=(−1)^2/271=−71. (14) The character of 𝐹₁ is the Kronecker symbol 𝜒(𝑑)=−71𝑑forℤ𝑑∈71ℤ^𝑥. (15)

Similarly, for the quadratic form Φ₁=2𝑥²₁−𝑥₁𝑥₂+9𝑥²₂, the determinant and a cofactor are 𝐷=71,𝐴₂₁=−1. (16) So, 𝛿=1,𝑁=𝐷=79,Δ=(−1)^2/271=−71, and the character of Φ₁ is (15).

For the quadratic form Ψ₁=3𝑥²₁−𝑥₁𝑥₂+6𝑥²₂, the determinant and a cofactor are 𝐷=71,𝐴₂₁=−1. (17) So, 𝛿=1,𝑁=𝐷=71,Δ=(−1)^2/271=−71, and the character of Ψ₁ is (15).

For the quadratic form Λ₁=4𝑥²₁−3𝑥₁𝑥₂+5𝑥²₂, the determinant and some cofactors are 𝐷=71,𝐴₁₂=−3,𝐴₁₁=10. (18) So, 𝛿=1,𝑁=𝐷=71,Δ=(−1)^2/271=−71, and the character of Ψ₁ is (15).

Consequently, 𝐹₁,Φ₁,Ψ₁,andΛ₁ are quadratic forms whose theta series are in 𝑀₁(Γ₀(71),(−71/𝑑)); hence, by Theorem 2,  𝐹₂,Φ₂,Ψ₂,Λ₂ and 𝐹₁⊕Φ₁,𝐹₁⊕Ψ₁,𝐹₁⊕Λ₁,Φ₁⊕Ψ₁,Φ₁⊕Λ₁,Ψ₁⊕Λ₁ (19) are quadratic forms whose theta series are in 𝑀₂(Γ₀(71)). Moreover, the theta series of quadratic forms in (4) are in 𝑀₄(Γ₀(71)), and the theta series of quadratic forms in (5) are in 𝑀₆(Γ₀(71)).

We have the following important theorem for the Eisenstein part of theta series associated with the quadratic form.

Theorem 3. Let 𝑄 be a positive definite form of 8 or 12 variables whose theta series Θ_𝑄 are in 𝑀₄(Γ₀(71)) and 𝑀₆(Γ₀(71)) respectively. Then, the Eisenstein parts of Θ_𝑄 are 𝐸𝑞;𝑄=1+^∞_𝑛=1𝛼₁𝜎₃(𝑛)𝑞^𝑛+𝛽₁𝜎₃(𝑛)𝑞^71𝑛𝑓𝑜𝑟8𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠,𝐸𝑞;𝑄=1+^∞_𝑛=1𝛼₂𝜎₅(𝑛)𝑞^𝑛+𝛽₂𝜎₅(𝑛)𝑞^71𝑛𝑓𝑜𝑟12𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠, (20)(21) respectively, where 𝜌₄=3!(2𝜋)⁴𝜁(4)=1240,𝜌₆=−5!(2𝜋)⁶𝜁(6)1=−,𝛼504₁=24071²−171⁴=−11202521,𝛼₂=50471³+171⁶=−136,𝛽25565₁=24071⁴−71²71⁴−1=71²120,𝛽2521₂=−50471⁶+71³71⁶−1=−12884796,𝐸25565_𝐹4𝑞=⋯=𝐸_{𝐹1⊕Φ1⊕Ψ1⊕Λ1}𝑞=1+1202521^∞_𝑛=1𝑞^𝑛+71²𝑞^71𝑛𝜎₃(𝑛)=1+1202521^∞_𝑛=1𝜎^∗₃(𝑛)𝑞^𝑛=1+120𝑞+2521120⋅9𝑞2521²+120⋅28𝑞2521³+120⋅73𝑞2521⁴+120⋅126𝑞2521⁵+120⋅252𝑞2521⁶+120⋅344𝑞2521⁷+120⋅585𝑞2521⁸+120⋅757𝑞2521⁹+120⋅1134𝑞2521¹⁰+120⋅1332𝑞2521¹¹+120⋅2044𝑞2521¹²+120⋅2198𝑞2521¹³+120⋅3096𝑞2521¹⁴+120⋅3528𝑞2521¹⁵+120⋅4681𝑞2521¹⁶+120⋅4914𝑞2521¹⁷𝜎+⋯,_∗3(𝑛)=𝜎₃(𝑛)if𝜎𝑛≥1,71∤𝑛₃(𝑛)+71²𝜎₃𝑛71if71∣𝑛,𝐸_𝐹6𝑞=⋯=𝐸_Λ6=1+3625565^∞_𝑛=1𝑞^𝑛−71³𝑞^71𝑛𝜎₅(𝑛)=1+3625565^∞_𝑛=1𝜎^∗₅(𝑛)𝑞^𝑛=1+3625565𝑞+1118𝑞25565²+8784𝑞25565³+38052𝑞25565⁴+112536𝑞25565⁵+289872𝑞25565⁶+605088𝑞25565⁷+243540𝑞5113⁸+2134548𝑞25565⁹+3713688𝑞25565¹⁰+5797872𝑞25565¹¹+9284688𝑞25565¹²[phone]𝑞25565¹³[phone]𝑞25565¹⁴[phone]𝑞25565¹⁵[phone]𝑞25565¹⁶[phone]𝑞25565¹⁷[phone]𝑞25565¹⁸[phone]𝑞5113¹⁹[phone]𝑞25565²⁰[phone]𝑞25565²¹[phone]𝑞25565²²[phone]𝑞25565²³[phone]𝑞5113²⁴[phone]𝑞25565²⁵[phone]𝑞25565²⁶[phone]𝑞25565²⁷[phone]𝑞5113²⁸[phone]𝑞5113²⁹𝜎+⋯,_∗5(𝑛)=𝜎₅(𝑛)if𝜎𝑛≥1,71∤𝑛₅(𝑛)−71³𝜎₅𝑛71if71∣𝑛. (22)(23)

Proof. See [1].

3. The Selection of Spherical Functions for Weight 4 and for Weight 6

First, we will be able to select 17 spherical functions such that the corresponding cusp forms become a basis of 𝑆₄(Γ₀(71)) since its dimension is 17; see [11].

For the quadratic form 2𝐹₂=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄=𝑥₁,𝑥₂,𝑥₃,𝑥₄21𝑥13621136¹𝑥₂𝑥₃𝑥₄, (24) the determinant and a cofactor are 𝐷=71²,𝐴₁₁=2556. By putting 2𝑘=4,𝑄=𝐹₂, and appropriate 𝑖,𝑗 in Theorem 2, we get 𝜑₁₁=𝑥²₁−14255671²2𝐹₂=𝑥²₁−18𝐹71², (25) which will be a spherical function of second order with respect to 𝐹₂. Similarly, 2𝐹₄=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄+2𝑥²₅+2𝑥₅𝑥₆+36𝑥²₆+2𝑥²₇+2𝑥₇𝑥₈+36𝑥²₈,𝐷=71⁴,𝐴₁₁𝜑=12884796,₁₁=𝑥²₁−181288479671⁴2𝐹₄=𝑥²₁−9𝐹71⁴,2Φ₂=4𝑥²₁−2𝑥₁𝑥₂+18𝑥²₂+4𝑥²₃−2𝑥₃𝑥₄+18𝑥²₄,𝐷=71²,𝐴₁₁=1278,𝐴₁₂𝜑=71,₁₁=𝑥²₁−14127871²2Φ₂=𝑥²₁−9Φ71²,𝜑₁₂=𝑥²₁−147171²2Φ₂=𝑥₁𝑥₂−1Φ142²,2𝐹₁⊕Φ₁=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+4𝑥²₃−2𝑥₃𝑥₄+18𝑥²₄,𝐷=71²,𝐴₁₁=2556,𝐴₁₂𝐴=−71,₃₃=1278,𝐴₃₄𝜑=71,₁₁=𝑥²₁−14255671²2𝐹₁⊕Φ₁=𝑥²₁−18𝐹71¹⊕Φ₁,𝜑₁₂=𝑥₁𝑥₂+147171²2𝐹₁⊕Φ₁=𝑥²₁+1𝐹142¹⊕Φ₁,𝜑₃₃=𝑥²₃−14127871²2𝐹₁⊕Φ₁=𝑥²₃−9𝐹71¹⊕Φ₁,𝜑₃₄=𝑥₃𝑥₄−147171²2𝐹₁⊕Φ₁=𝑥₃𝑥₄−1𝐹142¹⊕Φ₁,2Ψ₂=6𝑥²₁−2𝑥₁𝑥₂+12𝑥²₂+6𝑥²₃−2𝑥₃𝑥₄+12𝑥²₄,𝐷=71²,𝐴₁₁𝜑=852,₁₁=𝑥²₁−1485271²2Ψ₂=𝑥²₁−6Ψ71²,2𝐹₁⊕Ψ₁=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+6𝑥²₃−2𝑥₃𝑥₄+12𝑥²₄,𝐷=71²,𝐴₁₁=2556,𝐴₁₂𝐴=−71,₃₃=852,𝐴₄₄𝜑=426,₁₁=𝑥²₁−14255671²2𝐹₁⊕Ψ₁=𝑥²₁−18𝐹71¹⊕Ψ₁,𝜑₁₂=𝑥₁𝑥₂+147171²2𝐹₁⊕Ψ₁=𝑥₂𝑥₃+1𝐹142¹⊕Ψ₁,𝜑₃₃=𝑥²₃−1485271²2𝐹₁⊕Ψ₁=𝑥²₃−6𝐹71¹⊕Ψ₁,𝜑₄₄=𝑥²₄−1442671²2𝐹₁⊕Ψ₁=𝑥²₄−3𝐹71¹⊕Ψ₁2Φ₁⊕Ψ₁=4𝑥²₁−2𝑥₁𝑥₂+18𝑥²₂+6𝑥²₃−2𝑥₃𝑥₄+12𝑥²₄,𝐷=71²,𝐴₁₁=1278,𝐴₃₄𝜑=71,₁₁=𝑥²₁−14127871²2Φ₁⊕Ψ₁=𝑥²₁−9Φ71¹⊕Ψ₁,𝜑₃₄=𝑥₃𝑥₄−147171²2Φ₁⊕Ψ₁=𝑥₃𝑥₄−1Φ142¹⊕Ψ₁,2Λ₂=8𝑥²₁−6𝑥₁𝑥₂+10𝑥²₂+8𝑥²₃−6𝑥₃𝑥₄+10𝑥²₄,𝐷=71²,𝐴₂₂𝜑=568,₂₂=𝑥²₂−1456871²2Λ₂=𝑥²₂−4Λ71²,2𝐹₁⊕Λ₁=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+8𝑥²₃−6𝑥₃𝑥₄+10𝑥²₄,𝐷=71²,𝐴₁₁𝜑=2556,₁₁=𝑥²₁−14255671²2𝐹₁⊕Λ₁=𝑥²₁−18𝐹71¹⊕Λ₁,2Φ₁⊕Λ₁=4𝑥²₁−2𝑥₁𝑥₂+18𝑥²₂+8𝑥²₃−6𝑥₃𝑥₄+10𝑥²₄,𝐷=71²,𝐴₃₄𝜑=213,₃₄=𝑥₃𝑥₄−1421371²2Φ₁⊕Λ₁=𝑥₃𝑥₄−3Φ142¹⊕Λ₁. (26)

Similarly, we will be able to select 29 spherical functions such that the corresponding cusp forms become a basis of 𝑆₆(Γ₀(71)) since its dimension is 29; see [11], 2𝐹₄=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄+2𝑥²₅+2𝑥₅𝑥₆+36𝑥²₆+2𝑥²₇+2𝑥₇𝑥₈+36𝑥²₈,𝐷=71⁴,𝐴₁₁𝜑=12884796,₁₁=𝑥²₁−181288479671⁴2𝐹₄=𝑥²₁−9𝐹71⁴,2Φ₄=4𝑥²₁−2𝑥₁𝑥₂+18𝑥²₂+4𝑥²₃−2𝑥₃𝑥₄+18𝑥²₄+4𝑥²₅−2𝑥₅𝑥₆+18𝑥²₆+4𝑥²₇−2𝑥₇𝑥₈+18𝑥²₈,𝐷=71⁴,𝐴₁₁=6442398,𝐴₂₂𝜑=1431644,₁₁=𝑥²₁−18644239871⁴2Φ₄=𝑥²₁−9Φ142⁴,𝜑₂₂=𝑥²₂−18143164471⁴2Φ₄=𝑥²₂−1Φ71⁴,2𝐹₃⊕Φ₁=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄+2𝑥²₅+2𝑥₅𝑥₆+36𝑥²₆+4𝑥²₇−2𝑥₇𝑥₈+18𝑥²₈,𝐷=71⁴,𝐴₁₁=12884796,𝐴₂₂𝐴=715822,₇₇=6442398,𝐴₈₈𝜑=1431644,₁₁=𝑥²₁−181288479671⁴2𝐹₃⊕Φ₁=𝑥²₁−9𝐹71³⊕Φ₁,𝜑₂₂=𝑥²₂−1871582271⁴2𝐹₃⊕Φ₁=𝑥²₂−1𝐹142³⊕Φ₁,𝜑₇₇=𝑥²₇−18644239871⁴2𝐹₃⊕Φ₁=𝑥²₇−9𝐹142³⊕Φ₁,𝜑₈₈=𝑥²₈−18143164471⁴2𝐹₃⊕Φ₁=𝑥²₈−1𝐹71³⊕Φ₁,2𝐹₂⊕Φ₂=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄+4𝑥²₅−2𝑥₅𝑥₆+18𝑥²₆+4𝑥²₇−2𝑥₇𝑥₈+18𝑥²₈,𝐴₂₂=715822,𝐴₁₁=12884796,𝐴₅₅𝜑=6442398,₁₁=𝑥²₁−181288479671⁴2𝐹₂⊕Φ₂=𝑥²₁−9𝐹71²⊕Φ₂,𝜑₂₂=𝑥²₂−1871582271⁴2𝐹₂⊕Φ₂=𝑥²₂−1𝐹142²⊕Φ₂,𝜑₅₅=𝑥²₅−18644239871⁴2𝐹₂⊕Φ₂=𝑥²₅−9𝐹142²⊕Φ₂,2Ψ₄=6𝑥²₁−2𝑥₁𝑥₂+12𝑥²₂+6𝑥²₃−2𝑥₃𝑥₄+12𝑥²₄+6𝑥²₅−2𝑥₅𝑥₆+12𝑥²₆+6𝑥²₇−2𝑥₇𝑥₈+12𝑥²₈,𝐷=71⁴,𝐴₁₁=4294932,𝐴₁₂𝜑=357911,₁₁=𝑥²₁−18429493271⁴2Ψ₄=𝑥²₁−3Ψ71⁴,𝜑₁₂=𝑥₁𝑥₂−1835791171⁴2Ψ₄=𝑥₁𝑥₂−1Ψ284⁴,𝜑₂₂=𝑥²₂−18214746671⁴2Ψ₄=𝑥²₂−3Ψ142⁴,2𝐹₃⊕Ψ₁=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄+2𝑥²₅+2𝑥₅𝑥₆+36𝑥²₆+6𝑥²₇−2𝑥₇𝑥₈+12𝑥²₈,𝐴₁₁𝜑=12884796,₁₁=𝑥²₁−181288479671⁴2𝐹₃⊕Ψ₁=𝑥²₁−9𝐹71³⊕Ψ₁,2𝐹₂⊕Ψ₂=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄+6𝑥²₅−2𝑥₅𝑥₆+12𝑥²₆+6𝑥²₇−2𝑥₇𝑥₈+12𝑥²₈,𝐴₂₂=715822,𝐴₆₆𝜑=2147466,₂₂=𝑥²₂−1871582271⁴2𝐹₂⊕Ψ₂=𝑥²₂−1𝐹142²⊕Ψ₂,𝜑₆₆=𝑥²₆−18214746671⁴2𝐹₂⊕Ψ₂=𝑥²₆−3𝐹142²⊕Ψ₂,2Φ₃⊕Ψ₁=4𝑥²₁−2𝑥₁𝑥₂+18𝑥²₂+4𝑥²₃−2𝑥₃𝑥₄+18𝑥²₄+4𝑥²₅−2𝑥₅𝑥₆+18𝑥²₆+6𝑥²₃−2𝑥₃𝑥₄+12𝑥²₄,𝐷=71⁴,𝐴₁₁=6442398,𝐴₂₂𝜑=1431644,₁₁=𝑥²₁−18644239871⁴2Φ₃⊕Ψ₁=𝑥²₁−9Φ71³⊕Ψ₁,𝜑₂₂=𝑥²₂−18143164471⁴2Φ₃⊕Ψ₁=𝑥²₂−1Φ71³⊕Ψ₁,2Φ₁⊕Ψ₃=4𝑥²₁−2𝑥₁𝑥₂+18𝑥²₂+6𝑥²₃−2𝑥₃𝑥₄+12𝑥²₄+6𝑥²₅−2𝑥₅𝑥₆+12𝑥²₆+6𝑥²₃−2𝑥₃𝑥₄+12𝑥²₄,𝐷=71⁴,𝐴₂₂=1431644,𝐴₄₄𝜑=2147466,₂₂=𝑥²₂−18143164471⁴2Φ₁⊕Ψ₃=𝑥²₂−1Φ71¹⊕Ψ₃,𝜑₄₄=𝑥²₄−18214746671⁴2Φ₁⊕Ψ₃=𝑥²₄−3Φ142¹⊕Ψ₃,2𝐹₃⊕Λ₁=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄+2𝑥²₅+2𝑥₅𝑥₆+36𝑥²₆+8𝑥²₇−6𝑥₇𝑥₈+10𝑥²₈,𝐷=71⁴,𝐴₁₁=12884796,𝐴₂₂𝜑=715822,₁₁=𝑥²₁−181288479671⁴2𝐹₃⊕Λ₁=𝑥²₁−9𝐹71³⊕Λ₁,𝜑₂₂=𝑥²₂−1871582271⁴2𝐹₃⊕Λ₁=𝑥²₂−1𝐹142³⊕Λ₁,2Φ₁⊕Λ₃=4𝑥²₁−2𝑥₁𝑥₂+18𝑥²₂+8𝑥²₃−6𝑥₃𝑥₄+10𝑥²₄+8𝑥²₅−6𝑥₅𝑥₆+10𝑥²₆+8𝑥²₇−6𝑥₇𝑥₈+10𝑥²₈,𝐷=71⁴,𝐴₁₂=357911,𝐴₂₂𝐴=1431644,₄₄𝜑=2863288,₁₂=𝑥₁𝑥₂−1835791171⁴2Φ₁⊕Λ₃=𝑥₁𝑥₂−1Φ284¹⊕Λ₃,𝜑₂₂=𝑥²₂−18143164471⁴2Φ₁⊕Λ₃=𝑥²₂−1Φ71¹⊕Λ₃,𝜑₄₄=𝑥²₄−18286328871⁴2Φ₁⊕Λ₃=𝑥²₄−2Φ71¹⊕Λ₃,2Ψ₃⊕Λ₁=6𝑥²₁−2𝑥₁𝑥₂+12𝑥²₂+6𝑥²₃−2𝑥₃𝑥₄+12𝑥²₄+6𝑥²₅−2𝑥₅𝑥₆+12𝑥²₆+8𝑥²₇−6𝑥₇𝑥₈+10𝑥²₈,𝐷=71⁴,𝐴₂₂𝜑=2147466,₂₂=𝑥²₂−18214746671⁴2Ψ₃⊕Λ₁=𝑥²₂−3Ψ142³⊕Λ₁,2𝐹₂⊕Φ₁⊕Ψ₁=2𝑥²₁+2𝑥₁𝑥₂+36𝑥²₂+2𝑥²₃+2𝑥₃𝑥₄+36𝑥²₄+4𝑥²₅−2𝑥₅𝑥₆+18𝑥²₆+6𝑥²₇−2𝑥₇𝑥₈+12𝑥²₈,𝐷=71⁴,𝐴₅₅=6442398,𝐴₅₆𝐴=357911,₇₈𝜑=357911,₅₅=𝑥²₅−18644239871⁴2𝐹₂⊕Φ₁⊕Ψ₁=𝑥²₅−9𝐹142²⊕Φ₁⊕Ψ₁,𝜑₅₆=𝑥₅𝑥₆−1835791171⁴2𝐹₂⊕Φ₁⊕Ψ₁=𝑥₅𝑥₆−1𝐹284²⊕Φ₁⊕Ψ₁,𝜑₇₈=𝑥₇𝑥₈−1835791171⁴2𝐹₂⊕Φ₁⊕Ψ₁=𝑥₇𝑥₈−1𝐹284²⊕Φ₁⊕Ψ₁. (27)

4. The Solutions of 𝑄=𝑛 and the Theta Series Associated with the Quadratic Forms

(1)𝐹₁=𝑥²₁+𝑥₁𝑥₂+18𝑥²₂=𝑛 has the following solutions:

Consider the following: 𝑛=1⟹𝑥₁=±1,𝑥₂=0,𝑛=4⟹𝑥₁=±2,𝑥₂=0,𝑛=9⟹𝑥₁=±3,𝑥₂=0,𝑛=16⟹𝑥₁=±4,𝑥₂=0. (28) There is no integral solutions for 𝑛=2,3,5,6,7,8,10,11,12,13,14,15,17.

Consider the following:

(2)Φ₁=2𝑥²₁−𝑥₁𝑥₂+9𝑥²₂=𝑛 has the following solutions: 𝑛=2⟹𝑥₁=±1,𝑥₂=0,𝑛=8⟹𝑥₁=±2,𝑥₂=0,𝑛=9⟹𝑥₁=0,𝑥₂=±1,𝑛=10⟹𝑥₁=1,𝑥₂=−1or𝑥₁=−1,𝑥₂=1,𝑛=12⟹𝑥₁=1,𝑥₂=1or𝑥₁=−1,𝑥₂=−1,𝑛=15⟹𝑥₁=2,𝑥₂=−1or𝑥₁=−2,𝑥₂=1. (29) There is no integral solutions for 𝑛=1,3,4,5,6,7,11,13,14,16,17.

Consider the following:

(3)Ψ₁=3𝑥²₁−𝑥₁𝑥₂+6𝑥²₂=𝑛 has the following solutions: 𝑛=3⟹𝑥₁=±1,𝑥₂=0,𝑛=6⟹𝑥₁=0,𝑥₂=±1,𝑛=8⟹𝑥₁=1,𝑥₂=−1,or𝑥₁=−1,𝑥₂=1,𝑛=10⟹𝑥₁=1,𝑥₂=1,or𝑥₁=−1,𝑥₂=−1,𝑛=12⟹𝑥₁=±2,𝑥₂=0,𝑛=16⟹𝑥₁=−2,𝑥₂=1,𝑥₁=2,𝑥₂=−1. (30) There is no integral solutions for 𝑛=1,2,4,5,7,9,11,13,14,15,17.

(4)Λ₁=4𝑥²₁−3𝑥₁𝑥₂+5𝑥²₂ has the following solutions:

Consider the following: 𝑛=4⟹𝑥₁=±1,𝑥₂=0,𝑛=5⟹𝑥₁=0,𝑥₂=±1,𝑛=6⟹𝑥₁=1,𝑥₂=−1,or𝑥₁=−1,𝑥₂=1,𝑛=12⟹𝑥₁=1,𝑥₂=1,or𝑥₁=−1,𝑥₂=−1,𝑛=15⟹𝑥₁=2,𝑥₂=1,or𝑥₁=−2,𝑥₂=1,𝑛=16⟹𝑥₁=±2,𝑥₂=0. (31) There is no integral solutions for 𝑛=1,2,3,7,8,9,10,11,13,14,17.

Hence, for quadratic forms of 8 variables, we have Θ_𝐹1𝑞=1+2𝑞+2𝑞⁴+2𝑞⁹+2𝑞¹⁶+4𝑞¹⁸+4𝑞²⁰+4𝑞²⁴+2𝑞²⁵Θ+⋯,_Φ1𝑞=1+2𝑞²+2𝑞⁸+2𝑞⁹+2𝑞¹⁰+2𝑞¹²+2𝑞¹⁵+2𝑞¹⁸+2𝑞¹⁹+2𝑞²⁴Θ+⋯,_Ψ1𝑞=1+2𝑞³+2𝑞⁶+2𝑞⁸+2𝑞¹⁰+2𝑞¹²+2𝑞¹⁶+2𝑞²⁰+2𝑞²⁴+2𝑞²⁵+2𝑞²⁷+2𝑞²⁹Θ+⋯,_Λ1𝑞=1+2𝑞⁴+2𝑞⁵+2𝑞⁶+2𝑞¹²+2𝑞¹⁵+2𝑞¹⁶+2𝑞¹⁸+2𝑞²⁰+2𝑞²⁴+2𝑞²⁷Θ+⋯,_𝐹4𝑞=1+8𝑞+24𝑞²+32𝑞³+24𝑞⁴+48𝑞⁵+96𝑞⁶+64𝑞⁷+24𝑞⁸+104𝑞⁹+144𝑞¹⁰+96𝑞¹¹+96𝑞¹²+112𝑞¹³+192𝑞¹⁴+192𝑞¹⁵+24𝑞¹⁶+144𝑞¹⁷Θ+⋯,_Φ4𝑞=1+8𝑞²+24𝑞⁴+32𝑞⁶+24𝑞⁸+8𝑞⁹+56𝑞¹⁰+48𝑞¹¹+152𝑞¹²+96𝑞¹³+208𝑞¹⁴+72𝑞¹⁵+184𝑞¹⁶+96𝑞¹⁷Θ+⋯,_Ψ4𝑞=1+8𝑞³+32𝑞⁶+8𝑞⁸+80𝑞⁹+8𝑞¹⁰+48𝑞¹¹+144𝑞¹²+48𝑞¹³+144𝑞¹⁴+208𝑞¹⁵+176𝑞¹⁶+256𝑞¹⁷Θ+⋯,_Λ4𝑞=1+8𝑞⁴+8𝑞⁵+8𝑞⁶+24𝑞⁸+48𝑞⁹+72𝑞¹⁰+48𝑞¹¹+64𝑞¹²+96𝑞¹³+192𝑞¹⁴+232𝑞¹⁵+264𝑞¹⁶+208𝑞¹⁷Θ+⋯,_𝐹3⊕Φ1𝑞=1+6𝑞+14𝑞²+20𝑞³+30𝑞⁴+40𝑞⁵+36𝑞⁶+48𝑞⁷+62𝑞⁸+44𝑞⁹+86𝑞¹⁰+136𝑞¹¹+110𝑞¹²+160𝑞¹³+196𝑞¹⁴+162𝑞¹⁵+198𝑞¹⁶+204𝑞¹⁷Θ+⋯,_𝐹2⊕Φ2𝑞=1+4𝑞+8𝑞²+16𝑞³+24𝑞⁴+24𝑞⁵+32𝑞⁶+32𝑞⁷+24𝑞⁸+56𝑞⁹+68𝑞¹⁰+88𝑞¹¹+156𝑞¹²+152𝑞¹³+168𝑞¹⁴+196𝑞¹⁵+184𝑞¹⁶+216𝑞¹⁷Θ+⋯,_𝐹1⊕Φ3𝑞=1+2𝑞+6𝑞²+12𝑞³+14𝑞⁴+24𝑞⁵+20𝑞⁶+16𝑞¹⁷+30𝑞⁸+20𝑞⁹+58𝑞¹⁰+96𝑞¹¹+114𝑞¹²+168𝑞¹³+156𝑞¹⁴+166𝑞¹⁵+158𝑞¹⁶+180𝑞¹⁷Θ+⋯,_𝐹3⊕Ψ1𝑞=1+6𝑞+12𝑞²+10𝑞³+18𝑞⁴+48𝑞⁵+42𝑞⁶+24𝑞⁷+86𝑞⁸+106𝑞⁹+62𝑞¹⁰+124𝑞¹¹+154𝑞¹²+148𝑞¹³+204𝑞¹⁴+140𝑞¹⁵+188𝑞¹⁶+312𝑞¹⁷Θ+⋯,_𝐹2⊕Ψ2𝑞=1+4𝑞+4𝑞²+4𝑞³+20𝑞⁴+24𝑞⁵+8𝑞⁶+48𝑞⁷+72𝑞⁸+28𝑞⁹+92𝑞¹⁰+136𝑞¹¹+88𝑞¹²+176𝑞¹³+184𝑞¹⁴+168𝑞¹⁵+292𝑞¹⁶+248𝑞¹⁷Θ+⋯,_𝐹1⊕Ψ3𝑞=1+2𝑞+6𝑞³+14𝑞⁴+18𝑞⁶+48𝑞⁷+6𝑞⁸+46𝑞⁹+106𝑞¹⁰+36𝑞¹¹+114𝑞¹²+172𝑞¹³+108𝑞¹⁴+228𝑞¹⁵+248𝑞¹⁶+240𝑞¹⁷Θ+⋯,_Φ3⊕Ψ1𝑞=1+6𝑞²+2𝑞³+12𝑞⁴+12𝑞⁵+10𝑞⁶+24𝑞⁷+20𝑞⁸+22𝑞⁹+68𝑞¹⁰+36𝑞¹¹+120𝑞¹²+84𝑞¹³+160𝑞¹⁴+126𝑞¹⁵+198𝑞¹⁶+204𝑞¹⁷Θ+⋯,_Φ2⊕Ψ2𝑞=1+4𝑞²+4𝑞³+4𝑞⁴+16𝑞⁵+8𝑞⁶+16𝑞⁷+40𝑞⁸+12𝑞⁹+64𝑞¹⁰+64𝑞¹¹+68𝑞¹²+120𝑞¹³+128𝑞¹⁴+156𝑞¹⁵+196𝑞¹⁶+224𝑞¹⁷Θ+⋯,_Φ1⊕Ψ3𝑞=1+2𝑞²+6𝑞³+12𝑞⁵+18𝑞⁶+44𝑞⁸+34𝑞⁹+20𝑞¹⁰+100𝑞¹¹+68𝑞¹²+84𝑞¹³+168𝑞¹⁴+146𝑞¹⁵+210𝑞¹⁶+220𝑞¹⁷Θ+⋯,_𝐹3⊕Λ1𝑞=1+6𝑞+12𝑞²+8𝑞³+8𝑞⁴+38𝑞⁵+62𝑞⁶+52𝑞⁷+64𝑞⁸+106𝑞⁹+132𝑞¹⁰+120𝑞¹¹+82𝑞¹²+120𝑞¹³+204𝑞¹⁴+174𝑞¹⁵+144𝑞¹⁶+244𝑞¹⁷Θ+⋯,_𝐹2⊕Λ2𝑞=1+4𝑞+4𝑞²+8𝑞⁴+28𝑞⁵+36𝑞⁶+32𝑞⁷+40𝑞⁸+76𝑞⁹+116𝑞¹⁰+120𝑞¹¹+120𝑞¹²+168𝑞¹³+208𝑞¹⁴+180𝑞¹⁵+176𝑞¹⁶+224𝑞¹⁷Θ+⋯,_𝐹1⊕Λ3𝑞=1+2𝑞+8𝑞⁴+18𝑞⁵+18𝑞⁶+12𝑞⁷+24𝑞⁸+62𝑞⁹+96𝑞¹⁰+96𝑞¹¹+98𝑞¹²+136𝑞¹³+180𝑞¹⁴+218𝑞¹⁵+256𝑞¹⁶+276𝑞¹⁷Θ+⋯,_Φ3⊕Λ1𝑞=1+6𝑞²+14𝑞⁴+2𝑞⁵+22𝑞⁶+12𝑞⁷+42𝑞⁸+30𝑞⁹+70𝑞¹⁰+40𝑞¹¹+84𝑞¹²+48𝑞¹³+144𝑞¹⁴+128𝑞¹⁵+278𝑞¹⁶+264𝑞¹⁷Θ+⋯,_Φ2⊕Λ2𝑞=1+4𝑞²+8𝑞⁴+4𝑞⁵+20𝑞⁶+16𝑞⁷+40𝑞⁸+28𝑞⁹+56𝑞¹⁰+48𝑞¹¹+100𝑞¹²+96𝑞¹³+168𝑞¹⁴+136𝑞¹⁵+192𝑞¹⁶+168𝑞¹⁷Θ+⋯,_Φ1⊕Λ3𝑞=1+2𝑞²+6𝑞⁴+6𝑞⁵+18𝑞⁶+12𝑞⁷+26𝑞⁸+26𝑞⁹+62𝑞¹⁰+72𝑞¹¹+112𝑞¹²+96𝑞¹³+136𝑞¹⁴+136𝑞¹⁵+222𝑞¹⁶+256𝑞¹⁷Θ+⋯,_Ψ3⊕Λ1𝑞=1+6𝑞³+2𝑞⁴+2𝑞⁵+20𝑞⁶+12𝑞⁷+18𝑞⁸+44𝑞⁹+42𝑞¹⁰+60𝑞¹¹+92𝑞¹²+100𝑞¹³+136𝑞¹⁴+186𝑞¹⁵+212𝑞¹⁶+228𝑞¹⁷Θ+⋯,_Ψ2⊕Λ2𝑞=1+4𝑞³+4𝑞⁴+4𝑞⁵+12𝑞⁶+16𝑞⁷+24𝑞⁸+32𝑞⁹+48𝑞¹⁰+64q¹¹+96𝑞¹²+104𝑞¹³+136𝑞¹⁴+188𝑞¹⁵+220𝑞¹⁶+232𝑞¹⁷Θ+⋯,_Ψ1⊕Λ3𝑞=1+2𝑞³+6𝑞⁴+6𝑞⁵+8𝑞⁶+12𝑞⁷+26𝑞⁸+36𝑞⁹+50𝑞¹⁰+100𝑞¹²+108𝑞¹³+144𝑞¹⁴+174𝑞¹⁵+248𝑞¹⁶+252𝑞¹⁷Θ+⋯,_{𝐹2⊕Φ1⊕Ψ1}𝑞=1+4𝑞+4𝑞²+4𝑞⁴+8𝑞⁵++4𝑞⁸+4𝑞⁹+8𝑞¹⁰+8𝑞¹³+4𝑞¹⁶+8𝑞¹⁷Θ+⋯,_{𝐹2⊕Φ1⊕Λ1}𝑞=1+4𝑞+6𝑞²+8𝑞³+14𝑞⁴+18𝑞⁵+30𝑞⁶+52𝑞⁷+58𝑞⁸+70𝑞⁹+90𝑞¹⁰+88𝑞¹¹+96𝑞¹²+120𝑞¹³+152𝑞¹⁴+196𝑞¹⁵+246𝑞¹⁶+280𝑞¹⁷Θ+⋯,_{𝐹2⊕Ψ1⊕Λ1}𝑞=1+4𝑞+4𝑞²+2𝑞³+14𝑞⁴+26𝑞⁵+20𝑞⁶+36𝑞⁷+66𝑞⁸+72𝑞⁹+86𝑞¹⁰+100𝑞¹¹+124𝑞¹²+172𝑞¹³+168𝑞¹⁴+170𝑞¹⁵+256𝑞¹⁶+300𝑞¹⁷Θ+⋯,_{𝐹1⊕Φ2⊕Ψ1}𝑞=1+2𝑞+4𝑞²+10𝑞³+10𝑞⁴+16𝑞⁵+26𝑞⁶+16𝑞⁷+38𝑞⁸+50𝑞⁹+42𝑞¹⁰+100𝑞¹¹+102𝑞¹²+116𝑞¹³+172𝑞¹⁴+168𝑞¹⁵+212𝑞¹⁶+224𝑞¹⁷Θ+⋯,_{𝐹1⊕Φ2⊕Λ1}𝑞=1+2𝑞+4𝑞²+8𝑞³+8𝑞⁴+14𝑞⁵+22𝑞⁶+28𝑞⁷+48𝑞⁸+58𝑞⁹+64𝑞¹⁰+72𝑞¹¹+78𝑞¹²+96𝑞¹³+132𝑞¹⁴+202𝑞¹⁵+248𝑞¹⁶+292𝑞¹⁷Θ+⋯,_{Ψ1⊕Φ2⊕Λ1}𝑞=1+4𝑞²+2𝑞³+6𝑞⁴+10𝑞⁵+12𝑞⁶+20𝑞⁷+34𝑞⁸+32𝑞⁹+58𝑞¹⁰+52𝑞¹¹+88𝑞¹²+76𝑞¹³+152𝑞¹⁴+142𝑞¹⁵+224𝑞¹⁶+236𝑞¹⁷Θ+⋯,_{𝐹1⊕Λ1⊕Ψ2}𝑞=1+2𝑞+4𝑞³+12𝑞⁴+6𝑞⁵+14𝑞⁶+36𝑞⁷+32𝑞⁸+46q⁹+72𝑞¹⁰+80𝑞¹¹+114𝑞¹²+136𝑞¹³+148𝑞¹⁴+206𝑞¹⁵+260𝑞¹⁶+252𝑞¹⁷Θ+⋯,_{𝐹1⊕Φ1⊕Ψ2}𝑞=1+2𝑞+2𝑞²+8𝑞³+10𝑞⁴+8𝑞⁵+28𝑞⁶+24𝑞⁷+22𝑞⁸+72𝑞⁹+50𝑞¹⁰+64𝑞¹¹+142𝑞¹²+104𝑞¹³+148𝑞¹⁴+210𝑞¹⁵+210𝑞¹⁶+260𝑞¹⁷Θ+⋯,_{Λ1⊕Φ1⊕Ψ2}𝑞=1+2𝑞²+4𝑞³+2𝑞⁴+10𝑞⁵+14𝑞⁶+12𝑞⁷+34𝑞⁸+34𝑞⁹+46𝑞¹⁰+64𝑞¹¹+88𝑞¹²+96𝑞¹³+136𝑞¹⁴+172𝑞¹⁵+202𝑞¹⁶+240𝑞¹⁷Θ+⋯,_{𝐹1⊕Λ1⊕Ψ2}𝑞=1+2𝑞+4𝑞³+12𝑞⁴+6𝑞⁵+14𝑞⁶+36𝑞⁷+32𝑞⁸+46𝑞⁹+72𝑞¹⁰+80𝑞¹¹+114𝑞¹²+136𝑞¹³+148𝑞¹⁴+206𝑞¹⁵+260𝑞¹⁶+252𝑞¹⁷Θ+⋯,_{𝐹1⊕Φ1⊕Λ2}𝑞=1+2𝑞+2𝑞²+4𝑞³+6𝑞⁴+12𝑞⁵+24𝑞⁶+32𝑞⁷+38𝑞⁸+48𝑞⁹+66𝑞¹⁰+88𝑞¹¹+118𝑞¹²+144𝑞¹³+156𝑞¹⁴+174𝑞¹⁵+182𝑞¹⁶+212𝑞¹⁷Θ+⋯,_{𝐹1⊕Ψ1⊕Λ2}𝑞=1+2𝑞+2𝑞³+10𝑞⁴+12𝑞⁵+14𝑞⁶+24𝑞⁷+38𝑞⁸+54𝑞⁹+66𝑞¹⁰+84𝑞¹¹+122𝑞¹²+156𝑞¹³+156𝑞¹⁴+184𝑞¹⁵+252𝑞¹⁶+264𝑞¹⁷Θ+⋯,_{Φ1⊕Ψ1⊕Λ2}𝑞=1+2𝑞²+2𝑞³+4𝑞⁴+8𝑞⁵+14𝑞⁶+16𝑞⁷+28𝑞⁸+34𝑞⁹+52𝑞¹⁰+60𝑞¹¹+100𝑞¹²+100𝑞¹³+152𝑞¹⁴+150𝑞¹⁵+206𝑞¹⁶+220𝑞¹⁷Θ+⋯,_{𝐹1⊕Φ1⊕Ψ1⊕Λ1}𝑞=1+2𝑞+2𝑞²+6𝑞³+8𝑞⁴+10𝑞⁵+24𝑞⁶+28𝑞⁷+36𝑞⁸+60𝑞⁹+64𝑞¹⁰+76𝑞¹¹+102𝑞¹²+124𝑞¹³+148𝑞¹⁴+188𝑞¹⁵+234𝑞¹⁶+256𝑞¹⁷+⋯. (32) For quadratic forms of 12 variables, we have Θ_𝐹6𝑞=1+12𝑞+60𝑞²+160𝑞³+252𝑞⁴+312𝑞⁵+544𝑞⁶+960𝑞⁷+1020𝑞⁸+876𝑞⁹+1560𝑞¹⁰+2400𝑞¹¹+2080𝑞¹²+2040𝑞¹³+3264𝑞¹⁴+4160𝑞¹⁵+4092𝑞¹⁶+3480𝑞¹⁷+4404𝑞¹⁸+7440𝑞¹⁹+7536𝑞²⁰+6768𝑞²¹+11280𝑞²²+15168𝑞²³+16168𝑞²⁴+18420𝑞²⁵+21720𝑞²⁶+28720𝑞²⁷+32880𝑞²⁸+32232𝑞²⁹Θ+⋯,_Φ6𝑞=1+12𝑞²+60𝑞⁴+160𝑞⁶+252𝑞⁸+12𝑞⁹+324𝑞¹⁰+120𝑞¹¹+676𝑞¹²+480𝑞¹³+1560𝑞¹⁴+972𝑞¹⁵+2460𝑞¹⁶+1200𝑞¹⁷+2976𝑞¹⁸+1956𝑞¹⁹+4524𝑞²⁰+5040𝑞²¹+8664𝑞²²+9240𝑞²³+13312𝑞²⁴+11544𝑞²⁵+16080𝑞²⁶+14920𝑞²⁷+22224𝑞²⁸+25944𝑞²⁹Θ+⋯,_Ψ6𝑞=1+12𝑞³+72𝑞⁶+12𝑞⁸+280𝑞⁹+12𝑞¹⁰+120𝑞¹¹+792𝑞¹²+120𝑞¹³+600𝑞¹⁴+1752𝑞¹⁵+672𝑞¹⁶+1920𝑞¹⁷+3344𝑞¹⁸+2520𝑞¹⁹+4512𝑞²⁰+6144𝑞²¹+6960𝑞²²+8664𝑞²³+11632𝑞²⁴+14796𝑞²⁵+15600𝑞²⁶+21476𝑞²⁷+26280𝑞²⁸+27852𝑞²⁹Θ+⋯,_Λ6𝑞=1+12𝑞⁴+12𝑞⁵+12𝑞⁶+60𝑞⁸+120𝑞⁹+180𝑞¹⁰+120𝑞¹¹+232𝑞¹²+480𝑞¹³+960𝑞¹⁴+1132𝑞¹⁵+1332𝑞¹⁶+1560𝑞¹⁷+2692𝑞¹⁸+3960𝑞¹⁹+5484𝑞²⁰+6000𝑞²¹+6960𝑞²²+8280𝑞²³+12136𝑞²⁴+15576𝑞²⁵+18744𝑞²⁶+19172𝑞²⁷+22320𝑞²⁸+26904𝑞²⁹Θ+⋯,_𝐹5⊕Φ1𝑞=1+10𝑞+42𝑞²+100𝑞³+170𝑞⁴+272𝑞⁵+420𝑞⁶+544𝑞⁷+682𝑞⁸+912𝑞⁹+1062𝑞¹⁰+1320𝑞¹¹+1942𝑞¹²+2264𝑞¹³+2564𝑞¹⁴+3586𝑞¹⁵+4050𝑞¹⁶+4244𝑞¹⁷+5382𝑞¹⁸+6222𝑞¹⁹+7536𝑞²⁰+9540𝑞²¹+10924𝑞²²+13604𝑞²³+16206𝑞²⁴+17730𝑞²⁵+20900𝑞²⁶+24080𝑞²⁷+27964𝑞²⁸+33848𝑞²⁹Θ+⋯,_𝐹4⊕Φ2𝑞=1+8𝑞+28𝑞²+64𝑞³+124𝑞⁴+208𝑞⁵+288𝑞⁶+384𝑞⁷+508𝑞⁸+588𝑞⁹+764𝑞¹⁰+1096𝑞¹¹+1356𝑞¹²+1872𝑞¹³+2568𝑞¹⁴+2884𝑞¹⁵+3580𝑞¹⁶+23384𝑞¹⁷+4696𝑞¹⁸+5900𝑞¹⁹+7444𝑞²⁰+8912𝑞²¹+11144𝑞²²+13320𝑞²³+15584𝑞²⁴+17568𝑞²⁵+19936𝑞²⁶+23384𝑞²⁷+26800𝑞²⁸+32344𝑞²⁹Θ+⋯,_𝐹5⊕Ψ1𝑞=1+10𝑞+40𝑞²+82𝑞³+110𝑞⁴+192𝑞⁵+402𝑞⁶+520𝑞⁷+506𝑞⁸+910𝑞⁹+1462𝑞¹⁰+1364𝑞¹¹+1642𝑞¹²+2724𝑞¹³+3060𝑞¹⁴+3284𝑞¹⁵+4032𝑞¹⁶+4584𝑞¹⁷+6060𝑞¹⁸+6680𝑞¹⁹+6314𝑞²⁰+9344𝑞²¹+11700𝑞²²+11624𝑞²³+14422𝑞²⁴+17316𝑞²⁵+21064𝑞²⁶+25562𝑞²⁷+26304𝑞²⁸+33066𝑞²⁹Θ+⋯,_𝐹4⊕Ψ2𝑞=1+8𝑞+24𝑞²+36𝑞³+56𝑞⁴+144𝑞⁵+232𝑞⁶+224𝑞⁷+412𝑞⁸+784𝑞⁹+756𝑞¹⁰+936𝑞¹¹+1800𝑞¹²+1976𝑞¹³+2152𝑞¹⁴+3336𝑞¹⁵+3720𝑞¹⁶+4464𝑞¹⁷+5880𝑞¹⁸+5896𝑞¹⁹+7728𝑞²⁰+9984𝑞²¹+9872𝑞²²+12360𝑞²³+15104𝑞²⁴+16796𝑞²⁵+20608𝑞²⁶+22412𝑞²⁷+26456𝑞²⁸+34132𝑞²⁹Θ+⋯,_𝐹5⊕Λ1𝑞=1+10𝑞+40𝑞²+80𝑞³+92𝑞⁴+134𝑞⁵+342𝑞⁶+580𝑞⁷+620𝑞⁸+814𝑞⁹+1444𝑞¹⁰+1904𝑞¹¹+1922𝑞¹²+2120𝑞¹³+2900𝑞¹⁴+3862𝑞¹⁵+3972𝑞¹⁶+3844𝑞¹⁷+5502𝑞¹⁸+7320𝑞¹⁹+7118𝑞²⁰+7644𝑞²¹+10240𝑞²²+12584𝑞²³+14366𝑞²⁴+16938𝑞²⁵+21624𝑞²⁶+27222𝑞²⁷+29780𝑞²⁸+33104𝑞²⁹Θ+⋯,_𝐹4⊕Λ2𝑞=1+8𝑞+24𝑞²+32𝑞³+28𝑞⁴+84𝑞⁵+228𝑞⁶+320𝑞⁷+348𝑞⁸+560𝑞⁹+988𝑞¹⁰+1352𝑞¹¹+1544𝑞¹²+1904𝑞¹³+2784𝑞¹⁴+3620𝑞¹⁵+3748𝑞¹⁶+4088𝑞¹⁷+5396𝑞¹⁸+6504𝑞¹⁹+6996𝑞²⁰+8176𝑞²¹+10256𝑞²²+12360𝑞²³+14152𝑞²⁴+16384𝑞²⁵+20280𝑞²⁶+24012𝑞²⁷+26576𝑞²⁸+32632𝑞²⁹Θ+⋯,_{𝐹4⊕Φ1⊕Ψ1}𝑞=1+8𝑞+26𝑞²+50𝑞³+88𝑞⁴+164𝑞⁵+242𝑞⁶+320𝑞⁷+496𝑞⁸+650𝑞⁹+776𝑞¹⁰+1204𝑞¹¹+1532𝑞¹²+1700𝑞¹³+2552𝑞¹⁴+3130𝑞¹⁵+3282𝑞¹⁶+4516𝑞¹⁷+5358𝑞¹⁸+5942𝑞¹⁹+7726𝑞²⁰+8924𝑞²¹+10624𝑞²²+13060𝑞²³+14720𝑞²⁴+17478𝑞²⁵+20380𝑞²⁶+23554𝑞²⁷+27980𝑞²⁸+32174𝑞²⁹Θ+⋯,_{𝐹4⊕Φ1⊕Λ1}𝑞=1+8𝑞+26𝑞²+48𝑞³+74𝑞⁴+130𝑞⁵+214𝑞⁶+324𝑞⁷+510𝑞⁸+714𝑞⁹+914𝑞¹⁰+1264𝑞¹¹+1592𝑞¹²+1800𝑞¹³+2272𝑞¹⁴+2876𝑞¹⁵+3474𝑞¹⁶+4360𝑞¹⁷+5300𝑞¹⁸+6390𝑞¹⁹+7758𝑞²⁰+8824𝑞²¹+10076𝑞²²+12300𝑞²³+14696𝑞²⁴+17188𝑞²⁵+20988𝑞²⁶+24926𝑞²⁷+28496𝑞²⁸+33100𝑞²⁹Θ+⋯,_{𝐹4⊕Ψ1⊕Λ1}𝑞=1+8𝑞+24𝑞²+34𝑞³+42𝑞⁴+114𝑞⁵+228𝑞⁶+260𝑞⁷+366𝑞⁸+716𝑞⁹+966𝑞¹⁰+1092𝑞¹¹+1516𝑞¹²+2068𝑞¹³+2600𝑞¹⁴+3082𝑞¹⁵+3492𝑞¹⁶+4660𝑞¹⁷+5958𝑞¹⁸+6148𝑞¹⁹+7116𝑞²⁰+9356𝑞²¹+10636𝑞²²+11720𝑞²³+14020𝑞²⁴+16930𝑞²⁵+20576𝑞²⁶+23724𝑞²⁷+26388𝑞²⁸+32762𝑞²⁹Θ+⋯,_𝐹3⊕Φ3𝑞=1+6𝑞+18𝑞²+44𝑞³+90𝑞⁴+144𝑞⁵+212𝑞⁶+288𝑞⁷+330𝑞⁸+424𝑞⁹+570𝑞¹⁰+720𝑞¹¹+1130𝑞¹²+1584𝑞¹³+1980𝑞¹⁴+2694𝑞¹⁵+3114𝑞¹⁶+3516𝑞¹⁷+4402𝑞¹⁸+5154𝑞¹⁹+6540𝑞²⁰+8436𝑞²¹+10596𝑞²²+12876𝑞²³+14726𝑞²⁴+17382𝑞²⁵+19212𝑞²⁶+21800𝑞²⁷+26940𝑞²⁸+31440𝑞²⁹Θ+⋯,_𝐹3⊕Ψ3𝑞=1+6𝑞+12𝑞²+14𝑞³+144𝑞⁴+96𝑞⁵+90𝑞⁶+144𝑞⁷+378𝑞⁸+386𝑞⁹+402𝑞¹⁰+1020𝑞¹¹+1170q¹²+1116𝑞¹³+2244𝑞¹⁴+2604𝑞¹⁵+2748𝑞¹⁶+4464𝑞¹⁷+4784𝑞¹⁸+5616𝑞¹⁹+8106𝑞²⁰+8360𝑞²¹+10116𝑞²²+13080𝑞²³+14246𝑞²⁴+17292𝑞²⁵+19848𝑞²⁶+22758𝑞²⁷+28176𝑞²⁸+31254𝑞²⁹Θ+⋯,_𝐹3⊕Λ3𝑞=1+6𝑞+12𝑞²+8𝑞³+12𝑞⁴+66𝑞⁵+138𝑞⁶+156𝑞⁷+180𝑞⁸+354𝑞⁹+672𝑞¹⁰+936𝑞¹¹+1090𝑞¹²+1464𝑞¹³+2220𝑞¹⁴+2898𝑞¹⁵+3300𝑞¹⁶+4020𝑞¹⁷+5214𝑞¹⁸+6288𝑞¹⁹+7146𝑞²⁰+8212𝑞²¹+9744𝑞²²+11496𝑞²³+13318𝑞²⁴+16182𝑞²⁵+20016𝑞²⁶+23034𝑞²⁷+26316𝑞²⁸+32352𝑞²⁹Θ+⋯,_{𝐹3⊕Φ2⊕Ψ1}𝑞=1+6𝑞+16𝑞²+34𝑞³+70𝑞⁴+112𝑞⁵+162𝑞⁶+256𝑞⁷+330𝑞⁸+422𝑞⁹+658𝑞¹⁰+812𝑞¹¹+1062𝑞¹²+1620𝑞¹³+1876𝑞¹⁴+2384𝑞¹⁵+3288𝑞¹⁶+3600𝑞¹⁷+4520𝑞¹⁸+5852𝑞¹⁹+6742𝑞²⁰+8424𝑞²¹+10316𝑞²²+12392𝑞²³+14446𝑞²⁴+16888𝑞²⁵+20360𝑞²⁶+22810𝑞²⁷+27144𝑞²⁸+32578𝑞²⁹Θ+⋯,_{𝐹3⊕Φ2⊕Λ1}𝑞=1+6𝑞+16𝑞²+32𝑞³+60𝑞⁴+94𝑞⁵+142𝑞⁶+236𝑞⁷+348𝑞⁸+494𝑞⁹+720𝑞¹⁰+912𝑞¹¹+1134𝑞¹²+1472𝑞¹³+1772𝑞¹⁴+2258𝑞¹⁵+3020𝑞¹⁶+3828𝑞¹⁷+4826𝑞¹⁸+6012𝑞¹⁹+7170𝑞²⁰+8316𝑞²¹+9824𝑞²²+11608𝑞²³+13814𝑞²⁴+17094𝑞²⁵+20472𝑞²⁶+24166𝑞²⁷+28492𝑞²⁸+32200𝑞²⁹Θ+⋯,_{𝐹3⊕Φ1⊕Ψ2}𝑞=1+6𝑞+14𝑞²+24𝑞³+54𝑞⁴+96𝑞⁵+124𝑞⁶+216𝑞⁷+338𝑞⁸+380𝑞⁹+626𝑞¹⁰+928𝑞¹¹+966𝑞¹²+1536𝑞¹³+2148𝑞¹⁴+2162𝑞¹⁵+3198𝑞¹⁶+4156𝑞¹⁷+4362𝑞¹⁸+5982𝑞¹⁹+7264𝑞²⁰+8268𝑞²¹+10476𝑞²²+12228𝑞²³+14486𝑞²⁴+17010𝑞²⁵+19988𝑞²⁶+23700𝑞²⁷+26876𝑞²⁸+31980𝑞²⁹Θ+⋯,_{𝐹3⊕Ψ2⊕Λ1}𝑞=1+6𝑞+12𝑞²+12𝑞³+32𝑞⁴+86𝑞⁵+102𝑞⁶+132𝑞⁷+316𝑞⁸+450𝑞⁹+504𝑞¹⁰+840𝑞¹¹+1202𝑞¹²+1464𝑞¹³+1996𝑞¹⁴+2486𝑞¹⁵+3192𝑞¹⁶+4284𝑞¹⁷+4834𝑞¹⁸+5776𝑞¹⁹+7686𝑞²⁰+8844𝑞²¹+9920𝑞²²+12304𝑞²³+14558𝑞²⁴+16810𝑞²⁵+20032𝑞²⁶+22658𝑞²⁷+26764𝑞²⁸+32596𝑞²⁹Θ+⋯,_{𝐹3⊕Φ1⊕Λ2}𝑞=1+6𝑞+14𝑞²+20𝑞³+34𝑞⁴+68𝑞⁵+120𝑞⁶+208𝑞⁷+322𝑞⁸+436𝑞⁹+626𝑞¹⁰+904𝑞¹¹+1198𝑞¹²+1576𝑞¹³+2060𝑞¹⁴+2590𝑞¹⁵+3186𝑞¹⁶+3820𝑞¹⁷+4542𝑞¹⁸+5614𝑞¹⁹+6948𝑞²⁰+8300𝑞²¹+10180𝑞²²+12476𝑞²³+14510𝑞²⁴+16918𝑞²⁵+19780𝑞²⁶+22876𝑞²⁷+27020𝑞²⁸+31976𝑞²⁹Θ+⋯,_{𝐹3⊕Ψ1⊕Λ2}𝑞=1+8𝑞+12𝑞²+10𝑞³+22𝑞⁴+76𝑞⁵+118𝑞⁶+136𝑞⁷+250𝑞⁸+442𝑞⁹+602𝑞¹⁰+796𝑞¹¹+1114𝑞¹²+1612𝑞¹³+2164𝑞¹⁴+2544𝑞¹⁵+3152𝑞¹⁶+4264𝑞¹⁷+5132𝑞¹⁸+5736𝑞¹⁹+7094𝑞²⁰+8864𝑞²¹+10308𝑞²²+11952𝑞²³+13958𝑞²⁴+16728𝑞²⁵+20008𝑞²⁶+22398𝑞²⁷+26128𝑞²⁸+32226𝑞²⁹Θ+⋯,_{𝐹3⊕Φ1⊕Ψ1⊕Λ1}𝑞=1+6𝑞+14𝑞²+22𝑞³+44𝑞⁴+82𝑞⁵+120𝑞⁶+204𝑞⁷+328𝑞⁸+432𝑞⁹+648𝑞¹⁰+924𝑞¹¹+1102𝑞¹²+1492𝑞¹³+2004𝑞¹⁴+2396𝑞¹⁵+3094𝑞¹⁶+3944𝑞¹⁷+4772𝑞¹⁸+5938𝑞¹⁹+7144𝑞²⁰+8352𝑞²¹+10136𝑞²²+12180𝑞²³+14034𝑞²⁴+16904𝑞²⁵+20396𝑞²⁶+23340𝑞²⁷+27600𝑞²⁸+32194𝑞²⁹Θ+⋯,_𝐹2⊕Φ4𝑞=1+4𝑞+12𝑞²+32𝑞³+60𝑞⁴+104𝑞⁵+160𝑞⁶+192𝑞⁷+252𝑞⁸+300𝑞⁹+352𝑞¹⁰+592𝑞¹¹+824𝑞¹²+1224𝑞¹³+1808𝑞¹⁴+2120𝑞¹⁵+2652𝑞¹⁶+3048𝑞¹⁷+3452𝑞¹⁸+4456𝑞¹⁹+5608𝑞²⁰+7696𝑞²¹+9696𝑞²²+11984𝑞²³+14504𝑞²⁴+15596𝑞²⁵+18312𝑞²⁶+21152𝑞²⁷+24528𝑞²⁸+31144𝑞²⁹Θ+⋯,_𝐹2⊕Ψ4𝑞=1+4𝑞+4𝑞²+8𝑞³+36𝑞⁴+40𝑞⁵+32𝑞⁶+160𝑞⁷+204𝑞⁸+116𝑞⁹+496𝑞¹⁰+688𝑞¹¹+432𝑞¹²+1272𝑞¹³+1712𝑞¹⁴+1360𝑞¹⁵+2900𝑞¹⁶+3496𝑞¹⁷+3436𝑞¹⁸+5856𝑞¹⁹+6528𝑞²⁰+7344𝑞²¹+10352𝑞²²+11664𝑞²³+13912𝑞²⁴+16740𝑞²⁵+19656𝑞²⁶+23688𝑞²⁷+26752𝑞²⁸+31264𝑞²⁹Θ+⋯,_𝐹2⊕Λ4𝑞=1+4𝑞+4𝑞²+12𝑞⁴+48𝑞⁵+72𝑞⁶+64𝑞⁷+92𝑞⁸+244𝑞⁹+464𝑞¹⁰+592𝑞¹¹+672𝑞¹²+1000𝑞¹³+1632𝑞¹⁴+2248𝑞¹⁵+2764𝑞¹⁶+3488𝑞¹⁷+4420𝑞¹⁸+5536𝑞¹⁹+6696𝑞²⁰+8048𝑞²¹+9648𝑞²²+11280𝑞²³+13192𝑞²⁴+15948𝑞²⁵+18840𝑞²⁶+21320𝑞²⁷+25168𝑞²⁸+31304𝑞²⁹Θ+⋯,_{𝐹2⊕Φ3⊕Λ1}𝑞=1+4𝑞+10𝑞²+26𝑞³+48𝑞⁴+76𝑞⁵+130𝑞⁶+168𝑞⁷+224𝑞⁸+346𝑞⁹+404𝑞¹⁰+604𝑞¹¹+912𝑞¹²+1076𝑞¹³+1608𝑞¹⁴+2070𝑞¹⁵+2386𝑞¹⁶+3260𝑞¹⁷+3858𝑞¹⁸+4770𝑞¹⁹+6114𝑞²⁰+7540𝑞²¹+9512𝑞²²+11124𝑞²³+13912𝑞²⁴+16258𝑞²⁵+18316𝑞²⁶+23010𝑞²⁷+26244𝑞²⁸+30766𝑞²⁹Θ+⋯,_{𝐹2⊕Φ3⊕Λ1}𝑞=1+4𝑞+10𝑞²+24𝑞³+42𝑞⁴+66𝑞⁵+110𝑞⁶+156𝑞⁷+238𝑞⁸+370𝑞⁹+494𝑞¹⁰+688𝑞¹¹+892𝑞¹²+1072𝑞¹³+1400𝑞¹⁴+1816𝑞¹⁵+2418𝑞¹⁶+3256𝑞¹⁷+4264𝑞¹⁸+5314𝑞¹⁹+6314𝑞²⁰+7576𝑞²¹+8764𝑞²²+10444𝑞²³+13056𝑞²⁴+15928𝑞²⁵+19692𝑞²⁶+23822𝑞²⁷+27672𝑞²⁸+31628𝑞²⁹Θ+⋯,_{𝐹2⊕Φ1⊕Ψ3}𝑞=1+4𝑞+6𝑞²+14𝑞³+36𝑞⁴+44𝑞⁵+74𝑞⁶+160𝑞⁷+168𝑞⁸+262𝑞⁹+516𝑞¹⁰+492𝑞¹¹+764𝑞¹²+1348𝑞¹³+1248𝑞¹⁴+1898𝑞¹⁵+2910𝑞¹⁶+2844𝑞¹⁷+4098𝑞¹⁸+5478𝑞¹⁹+5982𝑞²⁰+7868𝑞²¹+9648𝑞²²+11500𝑞²³+13616𝑞²⁴+16454𝑞²⁵+20004𝑞²⁶+22230𝑞²⁷+27188𝑞²⁸+32178𝑞²⁹Θ+⋯,_{𝐹2⊕Ψ3⊕Λ1}𝑞=1+4𝑞+4𝑞²+6𝑞³+30𝑞⁴+42𝑞⁵+36𝑞⁶+124𝑞⁷+206𝑞⁸+192𝑞⁹+394𝑞¹⁰+636𝑞¹¹+700𝑞¹²+1100𝑞¹³+1528𝑞¹⁴+1850𝑞¹⁵+2680𝑞¹⁶+3324𝑞¹⁷+3978𝑞¹⁸+5484𝑞¹⁹+6676𝑞²⁰+7652𝑞²¹+9820𝑞²²+12008𝑞²³+13644𝑞²⁴+16738𝑞²⁵+19760𝑞²⁶+22488𝑞²⁷+27620𝑞²⁸+31358𝑞²⁹Θ+⋯,_{𝐹2⊕Φ1⊕Λ3}𝑞=1+4𝑞+6+8𝑞³+18𝑞⁴+38𝑞⁵+74𝑞⁶+124𝑞⁷+174𝑞⁸+254𝑞⁹+406𝑞¹⁰+624𝑞¹¹+888𝑞¹²+1200𝑞¹³+1568𝑞¹⁴+2032𝑞¹⁵+2634𝑞¹⁶+3328𝑞¹⁷+4204𝑞¹⁸+5286𝑞¹⁹+6346𝑞²⁰+7544𝑞²¹+9140𝑞²²+11052𝑞²³+13360𝑞²⁴+16232𝑞²⁵+19492𝑞²⁶+23202𝑞²⁷+27368𝑞²⁸+31548𝑞²⁹Θ+⋯,_{𝐹2⊕Ψ1⊕Λ3}𝑞=1+4𝑞+4𝑞²+2𝑞³+18𝑞⁴+46𝑞⁵+56𝑞⁶+76𝑞⁷+150𝑞⁸+264𝑞⁹+386𝑞¹⁰+524𝑞¹¹+772𝑞¹²+1172𝑞¹³+1568𝑞¹⁴+1950𝑞¹⁵+2636𝑞¹⁶+3572𝑞¹⁷+4374𝑞¹⁸+5284𝑞¹⁹+6592𝑞²⁰+8100𝑞²¹+9692𝑞²²+11320𝑞²³+13452𝑞²⁴+16454𝑞²⁵+19368𝑞²⁶+22184𝑞²⁷+26476𝑞²⁸+31474𝑞²⁹Θ+⋯,_{𝐹2⊕Φ2⊕Ψ1⊕Λ1}𝑞=1+4𝑞+8𝑞²+18𝑞³+34𝑞⁴+50𝑞⁵+92𝑞⁶+148𝑞⁷+206𝑞⁸+340𝑞⁹+474𝑞¹⁰+612𝑞¹¹+888𝑞¹²+1124𝑞¹³+1432𝑞¹⁴+1966𝑞¹⁵+2500𝑞¹⁶+3188𝑞¹⁷+4122𝑞¹⁸+5168𝑞¹⁹+6232𝑞²⁰+7636𝑞²¹+9340𝑞²²+10976𝑞²³+13500𝑞²⁴+16310𝑞²⁵+19176𝑞²⁶+23204𝑞²⁷+27108𝑞²⁸+31322𝑞²⁹Θ+⋯,_{𝐹2⊕Φ1⊕Ψ2⊕Λ1}𝑞=1+4𝑞+6𝑞²+12𝑞³+30𝑞⁴+42𝑞⁵+70𝑞⁶+140𝑞⁷+182𝑞⁸+278𝑞⁹+470𝑞¹⁰+568𝑞¹¹+800𝑞¹²+1200𝑞¹³+1440𝑞¹⁴+1924𝑞¹⁵+2630𝑞¹⁶+3216𝑞¹⁷+4104𝑞¹⁸+5238𝑞¹⁹+6326𝑞²⁰+7728𝑞²¹+9628𝑞²²+11324𝑞²³+13512𝑞²⁴+16716𝑞²⁵+19260𝑞²⁶+22754𝑞²⁷+27344𝑞²⁸+31160𝑞²⁹Θ+⋯,_{𝐹2⊕Φ1⊕Ψ1⊕Λ2}𝑞=1+4𝑞+6𝑞²+10𝑞³+24𝑞⁴+40𝑞⁵+70𝑞⁶+128𝑞⁷+184𝑞⁸+278𝑞⁹+436𝑞¹⁰+596𝑞¹¹+828𝑞¹²+1172𝑞¹³+1552𝑞¹⁴+2014𝑞¹⁵+2602𝑞¹⁶+3260𝑞¹⁷+4086𝑞¹⁸+5166𝑞¹⁹+6266𝑞²⁰+7700𝑞²¹+9640𝑞²²+11436𝑞²³+13616𝑞²⁴+16418𝑞²⁵+19180𝑞²⁶+22566𝑞²⁷+26780𝑞²⁸+31238𝑞²⁹Θ+⋯,_𝐹1⊕Φ5𝑞=1+2𝑞+10𝑞²+20𝑞³+42𝑞⁴+80𝑞⁵+100𝑞⁶+160𝑞⁷+170𝑞⁸+192𝑞⁹+302𝑞¹⁰+344q¹¹+670𝑞¹²+1000𝑞¹³+1364𝑞¹⁴+1930𝑞¹⁵+2082𝑞¹⁶+2500𝑞¹⁷+2814𝑞¹⁸+3414𝑞¹⁹+4984𝑞²⁰+6340𝑞²¹+9180𝑞²²+11060𝑞²³+12974𝑞²⁴+14922𝑞²⁵+16404𝑞²⁶+18560𝑞²⁷+24028𝑞²⁸+28360𝑞²⁹Θ+⋯,_𝐹1⊕Ψ5𝑞=1+2𝑞+10𝑞³+22𝑞⁴+50𝑞⁶+120𝑞⁷+10𝑞⁸+182𝑞⁹+430𝑞¹⁰+100𝑞¹¹+570𝑞¹²+1140𝑞¹³+500𝑞¹⁴+1572𝑞¹⁵+2456𝑞¹⁶+1720𝑞¹⁷+3684𝑞¹⁸+4744𝑞¹⁹+4634𝑞²⁰+7280𝑞²¹+8820𝑞²²+10280𝑞²³+12758𝑞²⁴+15892𝑞²⁵+19240𝑞²⁶+21490𝑞²⁷+27040𝑞²⁸+31650𝑞²⁹Θ+⋯,_𝐹1⊕Λ5𝑞=1+2𝑞+12𝑞⁴+30𝑞⁵+30𝑞⁶+20𝑞⁷+60𝑞⁸+182𝑞⁹+300𝑞¹⁰+320𝑞¹¹+370𝑞¹²+680𝑞¹³+1220𝑞¹⁴+1710𝑞¹⁵+2052𝑞¹⁶+2500𝑞¹⁷+3374𝑞¹⁸+4680𝑞¹⁹+6166𝑞²⁰+7564𝑞²¹+8800𝑞²²+10120𝑞²³+12398𝑞²⁴+15554𝑞²⁵+18664𝑞²⁶+21150𝑞²⁷+24740𝑞²⁸+29648𝑞²⁹Θ+⋯,_{𝐹1⊕Φ4⊕Ψ1}𝑞=1+2𝑞+8𝑞²+18𝑞³+30𝑞⁴+64𝑞⁵+82𝑞⁶+120𝑞⁷+186𝑞⁸+190𝑞⁹+334𝑞¹⁰+452𝑞¹¹+578𝑞¹²+996𝑞¹³+1172𝑞¹⁴+1612𝑞¹⁵+2128𝑞¹⁶+2424𝑞¹⁷+3300𝑞¹⁸+3904𝑞¹⁹+5234𝑞²⁰+6608𝑞²¹+8100𝑞²²+10696𝑞²³+12086𝑞²⁴+14732𝑞²⁵+18088𝑞²⁶+20058𝑞²⁷+25392𝑞²⁸+29946𝑞²⁹Θ+⋯,_{𝐹1⊕Φ4⊕Λ1}𝑞=1+2𝑞+8𝑞²+16𝑞³+28𝑞⁴+54𝑞⁵+70𝑞⁶+116𝑞⁷+172𝑞⁸+238𝑞⁹+380𝑞¹⁰+496𝑞¹¹+666𝑞¹²+856𝑞¹³+1060𝑞¹⁴+1422𝑞¹⁵+1908𝑞¹⁶+2692𝑞¹⁷+3574𝑞¹⁸+4544𝑞¹⁹+5638𝑞²⁰+6364𝑞²¹+7680𝑞²²+9160𝑞²³+11550𝑞²⁴+14658𝑞²⁵+18840𝑞²⁶+22390𝑞²⁷+26980𝑞²⁸+30016𝑞²⁹Θ+⋯,_{𝐹1⊕Φ1⊕Ψ4}𝑞=1+2𝑞+2𝑞²+12𝑞³+18𝑞⁴+16𝑞⁵+68𝑞⁶+80𝑞⁷+74𝑞⁸+264𝑞⁹+254𝑞¹⁰+280𝑞¹¹+790𝑞¹²+680𝑞¹³+900𝑞¹⁴+1890𝑞¹⁵+1674𝑞¹⁶+2404𝑞¹⁷+3806𝑞¹⁸+3918𝑞¹⁹+5344𝑞²⁰+6996𝑞²¹+8460𝑞²²+10148𝑞²³+12542𝑞²⁴+16162𝑞²⁵+17540𝑞²⁶+22008𝑞²⁷+27420𝑞²⁸+29472𝑞²⁹Θ+⋯,_{𝐹1⊕Ψ4⊕Λ1}𝑞=1+2𝑞+8𝑞³+20𝑞⁴+6𝑞⁵+38𝑞⁶+100𝑞⁷+60q⁸+150𝑞⁹+332𝑞¹⁰+288𝑞¹¹+514𝑞¹²+872𝑞¹³+932𝑞¹⁴+1462𝑞¹⁵+2060𝑞¹⁶+2388𝑞¹⁷+3414𝑞¹⁸+4520𝑞¹⁹+5262𝑞²⁰+6876𝑞²¹+8992𝑞²²+10296𝑞²³+12798𝑞²⁴+16170𝑞²⁵+18216𝑞²⁶+22478𝑞²⁷+27012𝑞²⁸+30104𝑞²⁹Θ+⋯,_{𝐹1⊕Φ1⊕Λ4}𝑞=1+2𝑞+2𝑞²+4𝑞³+10𝑞⁴+24𝑞⁵+44𝑞⁶+64q⁷+90𝑞⁸+152𝑞⁹+270𝑞¹⁰+424𝑞¹¹+598𝑞¹²+792𝑞¹³+1076𝑞¹⁴+1530𝑞¹⁵+2130𝑞¹⁶+2820𝑞¹⁷+3590𝑞¹⁸+4398𝑞¹⁹+5352𝑞²⁰+6676𝑞²¹+8476𝑞²²+10484𝑞²³+12718𝑞²⁴+14922𝑞²⁵+17732𝑞²⁶+21192𝑞²⁷+25820𝑞²⁸+30680𝑞²⁹Θ+⋯,_{𝐹1⊕Ψ1⊕Λ4}𝑞=1+2𝑞+2𝑞³+14𝑞⁴+24𝑞⁵+26𝑞⁶+40𝑞⁷+90𝑞⁸+166𝑞⁹+238𝑞¹⁰+324𝑞¹¹+506𝑞¹²+788𝑞¹³+1076𝑞¹⁴+1452𝑞¹⁵+2048𝑞¹⁶+2760𝑞¹⁷+3516𝑞¹⁸+4440𝑞¹⁹+5682𝑞²⁰+7200𝑞²¹+8820𝑞²²+10520𝑞²³+12822𝑞²⁴+15564𝑞²⁵+18280𝑞²⁶+214748𝑞²⁷+25824𝑞²⁸+30426𝑞²⁹Θ+⋯,_{𝐹1⊕Φ3⊕Ψ2}𝑞=1+2𝑞+6𝑞²+16𝑞³+22𝑞⁴+48𝑞⁵+76𝑞⁶+88𝑞⁷+178𝑞⁸+212𝑞⁹+278𝑞¹⁰+520𝑞¹¹+554𝑞¹²+856𝑞¹³+1276𝑞¹⁴+1382𝑞¹⁵+2102𝑞¹⁶+2596𝑞¹⁷+3174𝑞¹⁸+4274𝑞¹⁹+5164𝑞²⁰+6812𝑞²¹+8004𝑞²²+10220𝑞²³+12782𝑞²⁴+14118𝑞²⁵+18556𝑞²⁶+21516𝑞²⁷+24732𝑞²⁸+30996𝑞²⁹Θ+⋯,_{𝐹1⊕Φ3⊕Λ2}𝑞=1+2𝑞+6𝑞²+12𝑞³+18𝑞⁴+36𝑞⁵+56𝑞⁶+96𝑞⁷+162𝑞⁸+236𝑞⁹+342𝑞¹⁰+464𝑞¹¹+626𝑞¹²+848𝑞¹³+1172𝑞¹⁴+1586𝑞¹⁵+2042𝑞¹⁶+2612𝑞¹⁷+3226𝑞¹⁸+4002𝑞¹⁹+5216𝑞²⁰+6668𝑞²¹+8562𝑞²²+10580𝑞²³+12758𝑞²⁴+14714𝑞²⁵+17500𝑞²⁶+20356𝑞²⁷+24924𝑞²⁸+29824𝑞²⁹Θ+⋯,_{𝐹1⊕Φ3⊕Ψ1⊕Λ1}𝑞=1+2𝑞+6𝑞²+14𝑞³+20𝑞⁴+42𝑞⁵+64𝑞⁶+92𝑞⁷+168𝑞⁸+224𝑞⁹+332𝑞¹⁰+492𝑞¹¹+610𝑞¹²+852𝑞¹³+1124𝑞¹⁴+1472𝑞¹⁵+1982𝑞¹⁶+2616𝑞¹⁷+3440𝑞¹⁸+4246𝑞¹⁹+5452𝑞²⁰+6632𝑞²¹+7960𝑞²²+10028𝑞²³+12138𝑞²⁴+14756𝑞²⁵+18324𝑞²⁶+21716𝑞²⁷+25920𝑞²⁸+30282𝑞²⁹Θ+⋯,_{𝐹1⊕Φ2⊕Ψ3}𝑞=1+2𝑞+4𝑞²+14𝑞³+18𝑞⁴+32𝑞⁵+74𝑞⁶+72𝑞⁷+138𝑞⁸+250𝑞⁹+222𝑞¹⁰+468𝑞¹¹+670𝑞¹²+652𝑞¹³+1284𝑞¹⁴+1528𝑞¹⁵+1748𝑞¹⁶+2840𝑞¹⁷+3228𝑞¹⁸+4116𝑞¹⁹+5446𝑞²⁰+6704𝑞²¹+8380𝑞²²+9816𝑞²³+13086𝑞²⁴+14960𝑞²⁵+17416𝑞²⁶+22982𝑞²⁷+25208𝑞²⁸+30014𝑞²⁹Θ+⋯,_{𝐹1⊕Ψ3⊕Λ2}𝑞=1+2𝑞+6𝑞³+18𝑞⁴+12𝑞⁵+30𝑞⁶+80𝑞⁷+90𝑞⁸+142𝑞⁹+262𝑞¹⁰+356𝑞¹¹+530𝑞¹²+772𝑞¹³+1028𝑞¹⁴+1480𝑞¹⁵+2028𝑞¹⁶+2512𝑞¹⁷+3360𝑞¹⁸+4544𝑞¹⁹+5462𝑞²⁰+6856𝑞²¹+8852𝑞²²+10528𝑞²³+12934𝑞²⁴+15776𝑞²⁵+18376𝑞²⁶+22314𝑞²⁷+26656𝑞²⁸+30382𝑞²⁹Θ+⋯,_{𝐹1⊕Φ1⊕Ψ3⊕Λ1}𝑞=1+2𝑞+2𝑞²+10𝑞³+16𝑞⁴+18𝑞⁵+56𝑞⁶+76𝑞⁷+96𝑞⁸+220𝑞⁹+268𝑞¹⁰+356𝑞¹¹+646𝑞¹²+772𝑞¹³+1036𝑞¹⁴+1572𝑞¹⁵+1946𝑞¹⁶+2528𝑞¹⁷+3432𝑞¹⁸+4322𝑞¹⁹+5312𝑞²⁰+6912𝑞²¹+8520𝑞²²+10076𝑞²³+12938𝑞²⁴+15360𝑞²⁵+18036𝑞²⁶+22416𝑞²⁷+26024𝑞²⁸+30462𝑞²⁹Θ+⋯,_{𝐹1⊕Φ2⊕Λ3}𝑞=1+2𝑞+4𝑞²+8𝑞³+12𝑞⁴+26𝑞⁵+50𝑞⁶+84𝑞⁷+132𝑞⁸+194𝑞⁹+284𝑞¹⁰+424𝑞¹¹+638𝑞¹²+896𝑞¹³+1204𝑞¹⁴+1566𝑞¹⁵+1980𝑞¹⁶+2564𝑞¹⁷+3370𝑞¹⁸+4340𝑞¹⁹+5502𝑞²⁰+6724𝑞²¹+8112𝑞²²+9832𝑞²³+12302𝑞²⁴+15090𝑞²⁵+18688𝑞²⁶+22234𝑞²⁷+26292𝑞²⁸+30072𝑞²⁹Θ+⋯,_{𝐹1⊕Ψ2⊕Λ3}𝑞=1+2𝑞+4𝑞³+16𝑞⁴+18𝑞⁵+26𝑞⁶+60𝑞⁷+100𝑞⁸+150𝑞⁹+228𝑞¹⁰+352𝑞¹¹+546𝑞¹²+776𝑞¹³+1028𝑞¹⁴+1458𝑞¹⁵+2072𝑞¹⁶+2636𝑞¹⁷+3378𝑞¹⁸+4488𝑞¹⁹+5666𝑞²⁰+7012𝑞²¹+8656𝑞²²+10544𝑞²³+12982𝑞²⁴+15670𝑞²⁵+18360𝑞²⁶+21990𝑞²⁷+26468𝑞²⁸+30500𝑞²⁹Θ+⋯,_{𝐹1⊕Φ1⊕Ψ1⊕Λ3}𝑞=1+2𝑞+2𝑞²+6𝑞³+12𝑞⁴+22𝑞⁵+44𝑞⁶+68𝑞⁷+104𝑞⁸+172𝑞⁹+268𝑞¹⁰+396𝑞¹¹+582𝑞¹²+828𝑞¹³+1124𝑞¹⁴+1528𝑞¹⁵+2022𝑞¹⁶+2640𝑞¹⁷+3484𝑞¹⁸+4394𝑞¹⁹+5452𝑞²⁰+6832𝑞²¹+8432𝑞²²+10228𝑞²³+12570𝑞²⁴+15236𝑞²⁵+18324𝑞²⁶+21976q²⁷+26072𝑞²⁸+30426𝑞²⁹Θ+⋯,_{𝐹1⊕Φ2⊕Ψ2⊕Λ1}𝑞=1+2𝑞+4𝑞²+12𝑞³+16𝑞⁴+30𝑞⁵+62𝑞⁶+76𝑞⁷+140𝑞⁸+226𝑞⁹+276𝑞¹⁰+456𝑞¹¹+622𝑞¹²+784𝑞¹³+1156𝑞¹⁴+1506𝑞¹⁵+1952𝑞¹⁶+2604𝑞¹⁷+3406𝑞¹⁸+4244𝑞¹⁹+5322𝑞²⁰+6860𝑞²¹+8160𝑞²²+10112𝑞²³+12662𝑞²⁴+14838𝑞²⁵+18272𝑞²⁶+21938𝑞²⁷+25732𝑞²⁸+30492𝑞²⁹Θ+⋯,_{𝐹1⊕Φ2⊕Ψ1⊕Λ2}𝑞=1+2𝑞+4𝑞²+10𝑞³+14𝑞⁴+28𝑞⁵+54𝑞⁶+80𝑞⁷+138𝑞⁸+210𝑞⁹+294𝑞¹⁰+436𝑞¹¹+614𝑞¹²+844𝑞¹³+1156𝑞¹⁴+1564𝑞¹⁵+2008𝑞¹⁶+2576𝑞¹⁷+3352𝑞¹⁸+4172𝑞¹⁹+5346𝑞²⁰+6776𝑞²¹+8348𝑞²²+10336𝑞²³+12622𝑞²⁴+14988𝑞²⁵+18040𝑞²⁶+21518𝑞²⁷+25544𝑞²⁸+30234𝑞²⁹Θ+⋯,_Ψ1⊕Φ5𝑞=1+10𝑞²+2𝑞³+40𝑞⁴+20𝑞⁵+82𝑞⁶+80𝑞⁷+112𝑞⁸+170𝑞⁹+224𝑞¹⁰+260𝑞¹¹+612𝑞¹²+484𝑞¹³+1240𝑞¹⁴+1010𝑞¹⁵+1906𝑞¹⁶+1780𝑞¹⁷+2774𝑞¹⁸+2990𝑞¹⁹+4646𝑞²⁰+4940𝑞²¹+8304𝑞²²+7620𝑞²³+12672𝑞²⁴+11742𝑞²⁵+16924𝑞²⁶+17762𝑞²⁷+24220𝑞²⁸+26382𝑞²⁹Θ+⋯,_Φ5⊕Λ1𝑞=1+10𝑞²+42𝑞⁴+2𝑞⁵+102𝑞⁶+20𝑞⁷+190𝑞⁸+90𝑞⁹+362𝑞¹⁰+240𝑞¹¹+672𝑞¹²+440𝑞¹³+1104𝑞¹⁴+756𝑞¹⁵+1906𝑞¹⁶+1640𝑞¹⁷+3452𝑞¹⁸+3310𝑞¹⁹+5302𝑞²⁰+4920𝑞²¹+7164𝑞²²+6860𝑞²³+10952𝑞²⁴+12044q²⁵+18364𝑞²⁶+20062𝑞²⁷+26624𝑞²⁸+27020𝑞²⁹Θ+⋯,_Ψ2⊕Φ4𝑞=1+8𝑞²+4𝑞³+24𝑞⁴+32𝑞⁵+40𝑞⁶+96𝑞⁷+92𝑞⁸+144𝑞⁹+284𝑞¹⁰+216𝑞¹¹+576𝑞¹²+584𝑞¹³+888𝑞¹⁴+1264𝑞¹⁵+1512𝑞¹⁶+2048𝑞¹⁷+2720𝑞¹⁸+3168𝑞¹⁹+4808𝑞²⁰+4896𝑞²¹+7904𝑞²²+8152𝑞²³+11200𝑞²⁴+13300𝑞²⁵+16368𝑞²⁶+18620𝑞²⁷+25272𝑞²⁸+26420𝑞²⁹Θ+⋯,_Φ4⊕Λ2𝑞=1+8𝑞²+28𝑞⁴+4𝑞⁵+68𝑞⁶+32𝑞⁷+156𝑞⁸+112𝑞⁹+324𝑞¹⁰+248𝑞¹¹+576𝑞¹²+480𝑞¹³+1040𝑞¹⁴+972𝑞¹⁵+1892𝑞¹⁶+1800𝑞¹⁷+3004𝑞¹⁸+2944𝑞¹⁹+4684𝑞²⁰+5040𝑞²¹+7904𝑞²²+8632𝑞²³+12136𝑞²⁴+12536𝑞²⁵+16520𝑞²⁶+17308𝑞²⁷+23824𝑞²⁸+27000𝑞²⁹Θ+⋯,_{Φ4⊕Ψ1⊕Λ1}𝑞=1+8𝑞²+2𝑞³+26𝑞⁴+18𝑞⁵+52𝑞⁶+68𝑞⁷+110𝑞⁸+156𝑞⁹+270𝑞¹⁰+292𝑞¹¹+548𝑞¹²+532𝑞¹³+968𝑞¹⁴+978𝑞¹⁵+1732𝑞¹⁶+1876𝑞¹⁷+2990𝑞¹⁸+3324𝑞¹⁹+4852𝑞²⁰+5052𝑞²¹+7564𝑞²²+7704𝑞²³+11492𝑞²⁴+12426𝑞²⁵+17488𝑞²⁶+18940𝑞²⁷+25428𝑞²⁸+27194𝑞²⁹Θ+⋯,_Φ3⊕Ψ3𝑞=1+6𝑞²+6𝑞³+12𝑞⁴+36𝑞⁵+26𝑞⁶+72𝑞⁷+120𝑞⁸+86𝑞⁹+288𝑞¹⁰+276𝑞¹¹+384𝑞¹²+756𝑞¹³+720𝑞¹⁴+1174𝑞¹⁵+1662𝑞¹⁶+1812𝑞¹⁷+2870𝑞¹⁸+3186𝑞¹⁹+4674𝑞²⁰+5364𝑞²¹+7032𝑞²²+9276𝑞²³+10376𝑞²⁴+13458𝑞²⁵+17364𝑞²⁶+18230𝑞²⁷+25356𝑞²⁸+28098𝑞²⁹Θ+⋯,_Φ3⊕Λ3𝑞=1+6𝑞²+18𝑞⁴+6𝑞⁵+50𝑞⁶+36𝑞⁷+126𝑞⁸+102𝑞⁹+258𝑞¹⁰+240𝑞¹¹+524𝑞¹²+552𝑞¹³+1032𝑞¹⁴+1052𝑞¹⁵+1722𝑞¹⁶+1776𝑞¹⁷+2832𝑞¹⁸+3186𝑞¹⁹+4902𝑞²⁰+5416𝑞²¹+7572𝑞²²+8124𝑞²³+11160𝑞²⁴+12636𝑞²⁵+17604𝑞²⁶+19874𝑞²⁷+25728𝑞²⁸+27612𝑞²⁹Θ+⋯,_{Φ3⊕Ψ2⊕Λ1}𝑞=1+6𝑞²+4𝑞³+14𝑞⁴+26𝑞⁵+30𝑞⁶+68𝑞⁷+102𝑞⁸+126𝑞⁹+258𝑞¹⁰+280𝑞¹¹+468𝑞¹²+600𝑞¹³+856𝑞¹⁴+1104𝑞¹⁵+1574𝑞¹⁶+2016𝑞¹⁷+2732𝑞¹⁸+3362𝑞¹⁹+4706𝑞²⁰+5232𝑞²¹+7420𝑞²²+8476𝑞²³+11136𝑞²⁴+13072𝑞²⁵+17100𝑞²⁶+19154𝑞²⁷+24984𝑞²⁸+27784𝑞²⁹Θ+⋯,_{Φ3⊕Ψ1⊕Λ2}𝑞=1+6𝑞²+2𝑞³+16𝑞⁴+16𝑞⁵+38𝑞⁶+56𝑞⁷+104𝑞⁸+134𝑞⁹+240𝑞¹⁰+284𝑞¹¹+480𝑞¹²+564𝑞¹³+944𝑞¹⁴+1066𝑞¹⁵+1690𝑞¹⁶+1908𝑞¹⁷+2794𝑞¹⁸+3194𝑞¹⁹+4686𝑞²⁰+5276𝑞²¹+7632𝑞²²+8524𝑞²³+11608𝑞²⁴+12798𝑞²⁵+17036𝑞²⁶+18710𝑞²⁷+24676𝑞²⁸+27622𝑞²⁹Θ+⋯,_Φ2⊕Ψ4𝑞=1+4𝑞²+8𝑞³+4𝑞⁴+32𝑞⁵+32𝑞⁶+32𝑞⁷+140𝑞⁸+84𝑞⁹+180𝑞¹⁰+408𝑞¹¹+252𝑞¹²+656𝑞¹³+952𝑞¹⁴+820𝑞¹⁵+1748𝑞¹⁶+1968𝑞¹⁷+2480𝑞¹⁸+3548𝑞¹⁹+4356𝑞²⁰+5776𝑞²¹+6696𝑞²²+9240𝑞²³+11408𝑞²⁴+12272𝑞²⁵+18128𝑞²⁶+19632𝑞²⁷+23424𝑞²⁸+30192𝑞²⁹Θ+⋯,_Φ2⊕Λ4𝑞=1+4𝑞²+12𝑞⁴+8𝑞⁵+40𝑞⁶+32𝑞⁷+92𝑞⁸+84𝑞⁹+212𝑞¹⁰+248𝑞¹¹+492𝑞¹²+544𝑞¹³+904𝑞¹⁴+1004𝑞¹⁵+1644𝑞¹⁶+1984𝑞¹⁷+3016𝑞¹⁸+3420𝑞¹⁹+4684𝑞²⁰+5168𝑞²¹+7208𝑞²²+8568𝑞²³+11872𝑞²⁴+13496𝑞²⁵+17216𝑞²⁶+18832𝑞²⁷+24176𝑞²⁸+27960𝑞²⁹Θ+⋯,_Φ1⊕Ψ5𝑞=1+2𝑞²+10𝑞³+20𝑞⁵+50𝑞⁶+112𝑞⁸+162𝑞⁹+32𝑞¹⁰+420𝑞¹¹+412𝑞¹²+260𝑞¹³+1160𝑞¹⁴+954𝑞¹⁵+1130𝑞¹⁶+2484𝑞¹⁷+2342𝑞¹⁸+3222𝑞¹⁹+4670𝑞²⁰+5660𝑞²¹+7040𝑞²²+8564𝑞²³+12256𝑞²⁴+12934𝑞²⁵+16300𝑞²⁶+22314𝑞²⁷+23100𝑞²⁸+29110𝑞²⁹Θ+⋯,_Φ1⊕Λ5𝑞=1+2𝑞²+10𝑞⁴+10𝑞⁵+30𝑞⁶+20𝑞⁷+62𝑞⁸+82𝑞⁹+202𝑞¹⁰+240𝑞¹¹+392𝑞¹²+440𝑞¹³+800𝑞¹⁴+1092𝑞¹⁵+1730𝑞¹⁶+2040𝑞¹⁷+2772𝑞¹⁸+3222𝑞¹⁹+4662𝑞²⁰+5880𝑞²¹+8044𝑞²²+9100𝑞²³+11192𝑞²⁴+12492𝑞²⁵+16620𝑞²⁶+20134𝑞²⁷+25984𝑞²⁸+29004𝑞²⁹Θ+⋯,_{Φ1⊕Ψ4⊕Λ1}𝑞=1+2𝑞²+8𝑞³+2𝑞⁴+18𝑞⁵+38𝑞⁶+20𝑞⁷+94𝑞⁸+130𝑞⁹+122𝑞¹⁰+320𝑞¹¹+392𝑞¹²+472𝑞¹³+864𝑞¹⁴+1068𝑞¹⁵+1402𝑞¹⁶+2008𝑞¹⁷+2700𝑞¹⁸+3286𝑞¹⁹+4462𝑞²⁰+5864𝑞²¹+6908𝑞²²+9068𝑞²³+11560𝑞²⁴+13204𝑞²⁵+17276𝑞²⁶+20486𝑞²⁷+24368𝑞²⁸+29380𝑞²⁹Θ+⋯,_{Φ1⊕Ψ1⊕Λ4}𝑞=1+2𝑞²+2𝑞³+8𝑞⁴+12𝑞⁵+26𝑞⁶+32𝑞⁷+64𝑞⁸+98𝑞⁹+176𝑞¹⁰+244𝑞¹¹+396𝑞¹²+516𝑞¹³+808𝑞¹⁴+1058𝑞¹⁵+1570𝑞¹⁶+2004𝑞¹⁷+2830𝑞¹⁸+3398𝑞¹⁹+4630𝑞²⁰+5612𝑞²¹+7504𝑞²²+8916𝑞²³+11584𝑞²⁴+13406𝑞²⁵+17084𝑞²⁶+19786𝑞²⁷+24844𝑞²⁸+28638𝑞²⁹Θ+⋯,_{Φ1⊕Ψ3⊕Λ2}𝑞=1+2𝑞²+6𝑞³+4𝑞⁴+16𝑞⁵+30𝑞⁶+32𝑞⁷+80𝑞⁸+114𝑞⁹+160𝑞¹⁰+268𝑞¹¹+396𝑞¹²+532𝑞¹³+792𝑞¹⁴+1102𝑞¹⁵+1462𝑞¹⁶+1972𝑞¹⁷+2714𝑞¹⁸+3326𝑞¹⁹+4554𝑞²⁰+5716𝑞²¹+7160𝑞²²+9044𝑞²³+11440𝑞²⁴+13506𝑞²⁵+17108𝑞²⁶+20314𝑞²⁷+24644𝑞²⁸+29002𝑞²⁹Θ+⋯,_{Φ1⊕Ψ2⊕Λ3}𝑞=1+2𝑞²+4𝑞³+6𝑞⁴+14𝑞⁵+26𝑞⁶+36𝑞⁷+70𝑞⁸+106𝑞⁹+170𝑞¹⁰+248𝑞¹¹+400𝑞¹²+536𝑞¹³+800𝑞¹⁴+1080𝑞¹⁵+1502𝑞¹⁶+1992𝑞¹⁷+2736𝑞¹⁸+3398𝑞¹⁹+4594𝑞²⁰+5680𝑞²¹+7316𝑞²²+8892𝑞²³+11488𝑞²⁴+13520𝑞²⁵+17188𝑞²⁶+20134𝑞²⁷+24904𝑞²⁸+28744𝑞²⁹Θ+⋯,_Ψ5⊕Λ1𝑞=1+10𝑞³+2𝑞⁴+2𝑞⁵+52𝑞⁶+20𝑞⁷+30𝑞⁸+180𝑞⁹+110𝑞¹⁰+180𝑞¹¹+492𝑞¹²+420𝑞¹³+680𝑞¹⁴+1194𝑞¹⁵+1292𝑞¹⁶+1860𝑞¹⁷+2782𝑞¹⁸+3284𝑞¹⁹+4236𝑞²⁰+6044𝑞²¹+7180𝑞²²+8680𝑞²³+12052𝑞²⁴+13730𝑞²⁵+16800𝑞²⁶+21636𝑞²⁷+24500𝑞²⁸+29570𝑞²⁹Θ+⋯,_Ψ4⊕Λ2𝑞=1+8𝑞³+4𝑞⁴+4𝑞⁵+36𝑞⁶+32𝑞⁷+44𝑞⁸+120𝑞⁹+148𝑞¹⁰+216𝑞¹¹+376𝑞¹²+496𝑞¹³+720𝑞¹⁴+1076𝑞¹⁵+1404𝑞¹⁶+1864𝑞¹⁷+2732𝑞¹⁸+3384𝑞¹⁹+4324𝑞²⁰+5936𝑞²¹+7280𝑞²²+8984𝑞²³+11736𝑞²⁴+13872𝑞²⁵+17128𝑞²⁶+21092𝑞²⁷+24928𝑞²⁸+29296𝑞²⁹Θ+⋯,_Ψ3⊕Λ3𝑞=1+6𝑞³+6𝑞⁴+6𝑞⁵+24𝑞⁶+36𝑞⁷+54𝑞⁸+92𝑞⁹+150𝑞¹⁰+228𝑞¹¹+356𝑞¹²+492𝑞¹³+720𝑞¹⁴+1070𝑞¹⁵+1440𝑞¹⁶+1884𝑞¹⁷+2698𝑞¹⁸+3468𝑞¹⁹+4488𝑞²⁰+5788𝑞²¹+7404𝑞²²+9144𝑞²³+11652𝑞²⁴+13926𝑞²⁵+17208𝑞²⁶+20884𝑞²⁷+25116𝑞²⁸+29142𝑞²⁹Θ+⋯,_Ψ2⊕Λ4𝑞=1+4𝑞³+8𝑞⁴+18𝑞⁵+16𝑞⁶+32𝑞⁷+60𝑞⁸+88𝑞⁹+140𝑞¹⁰+216𝑞¹¹+360𝑞¹²+488𝑞¹³+712𝑞¹⁴+1040𝑞¹⁵+1496𝑞¹⁶+1936𝑞¹⁷+2664𝑞¹⁸+3480𝑞¹⁹+4648𝑞²⁰+5824𝑞²¹+7472𝑞²²+9176𝑞²³+11776𝑞²⁴+13940𝑞²⁵+17216𝑞²⁶+20516𝑞²⁷+25208𝑞²⁸+28980𝑞²⁹Θ+⋯,_Ψ1⊕Λ5𝑞=1+2𝑞³+10𝑞⁴+10𝑞⁵+12𝑞⁶+20𝑞⁷+62𝑞⁸+100𝑞⁹+142𝑞¹⁰+180𝑞¹¹+332𝑞¹²+500𝑞¹³+760𝑞¹⁴+1010𝑞¹⁵+1492𝑞¹⁶+1940𝑞¹⁷+2710𝑞¹⁸+3460𝑞¹⁹+4804𝑞²⁰+5980𝑞²¹+7660𝑞²²+9064𝑞²³+11732𝑞²⁴+13994𝑞²⁵+17424𝑞²⁶+20292𝑞²⁷+25028𝑞²⁸+28682𝑞²⁹Θ+⋯,_{Φ1⊕Ψ4⊕Λ1}𝑞=1+2𝑞²+8𝑞³+2𝑞⁴+18q⁵+38𝑞⁶+20𝑞⁷+94𝑞⁸+130𝑞⁹+122𝑞¹⁰+320𝑞¹¹+392𝑞¹²+472𝑞¹³+864𝑞¹⁴+1068𝑞¹⁵+1402𝑞¹⁶+2008𝑞¹⁷+2700𝑞¹⁸+3286𝑞¹⁹+4462𝑞²⁰+5864𝑞²¹+6908𝑞²²+9068𝑞²³+11560𝑞²⁴+13204𝑞²⁵+17276𝑞²⁶+20486𝑞²⁷+24368𝑞²⁸+29380𝑞²⁹+⋯. (33)

Now, we will easily determine some bases of 𝑆₄(Γ₀(71)),𝑆₆(Γ₀(71)).

Theorem 4. The following generalized 17 theta series Θ_𝐹2,𝜑11𝑞=^∞_{𝑛=1𝐹2=𝑛}𝑥²₁−18𝐹71²𝑞^𝑛,Θ_Φ2,𝜑11𝑞=^∞_{𝑛=1Φ2=𝑛}𝑥²₁−9Φ142²𝑞^𝑛,Θ_Φ2,𝜑12𝑞=^∞_{𝑛=1Φ2=𝑛}𝑥₁𝑥₂−1Φ142²𝑞^𝑛,Θ_{𝐹1⊕Φ1,𝜑11}𝑞=^∞_{𝑛=1Φ2=𝑛}𝑥²₁−18𝐹71¹⊕Φ₁𝑞^𝑛,Θ_{𝐹1⊕Φ1,𝜑12}𝑞=^∞_{𝑛=1𝐹1⊕Φ1=𝑛}𝑥₁𝑥₂+1𝐹142¹⊕Φ₁𝑞^𝑛,Θ_{𝐹1⊕Φ1,𝜑33}𝑞=^∞_{𝑛=1Φ2=𝑛}𝑥²₃−9𝐹71¹⊕Φ₁𝑞^𝑛,Θ_{𝐹1⊕Φ1,𝜑34}𝑞=^∞_{𝑛=1𝐹1⊕Φ1=𝑛}𝑥₃𝑥₄−1𝐹142¹⊕Φ₁𝑞^𝑛,Θ_Ψ2,𝜑11𝑞=^∞_{𝑛=1Ψ2=𝑛}𝑥²₁−6Ψ71²𝑞^𝑛,Θ_{𝐹1⊕Ψ1,𝜑11}𝑞=^∞_{𝑛=1𝐹1⊕Ψ1=𝑛}𝑥²₁−18𝐹71¹⊕Ψ₁𝑞^𝑛,Θ_{𝐹1⊕Ψ1,𝜑12}𝑞=^∞_{𝑛=1𝐹1⊕Ψ1=𝑛}𝑥₁𝑥₂+1𝐹142¹⊕Ψ₁𝑞^𝑛,Θ_{𝐹1⊕Ψ1,𝜑33}𝑞=^∞_{𝑛=1𝐹1⊕Ψ1=𝑛}𝑥²₃−6𝐹71¹⊕Ψ₁𝑞^𝑛,Θ_{𝐹1⊕Ψ1,𝜑44}𝑞=^∞_{𝑛=1𝐹1⊕Ψ1=𝑛}𝑥²₄−3𝐹71¹⊕Ψ₁𝑞^𝑛,Θ_{Φ1⊕Ψ1,𝜑11}𝑞=^∞_{𝑛=1Ψ2=𝑛}𝑥²₁−9Φ71¹⊕Ψ₁𝑞^𝑛,Θ_{Φ1⊕Ψ1,𝜑34}𝑞=^∞_{𝑛=1Ψ2=𝑛}𝑥₃𝑥₄−1Φ142¹⊕Ψ₁𝑞^𝑛,Θ_Λ2,𝜑22𝑞=171^∞_{𝑛=1Λ2=𝑛}71𝑥²₂−4Λ₂𝑞^𝑛,Θ_{𝐹1⊕Λ1,𝜑11}𝑞=^∞_{𝑛=1𝐹1⊕Λ1=𝑛}𝑥²₁−18𝐹71¹⊕Λ₁𝑞^𝑛,Θ_{Ψ1⊕Λ1,𝜑34}𝑞=∑^∞_𝑛=1∑_{Ψ1⊕Λ1=𝑛}𝑥₃𝑥₄−3Ψ142¹⊕Λ₁𝑞^𝑛, (34) are a basis of 𝑆₄(Γ₀(71)), and the following generalized theta series Θ_𝐹4,𝜑11𝑞=^∞_{𝑛=1𝐹4=𝑛}𝑥²₁−9𝐹71⁴𝑞^𝑛,Θ_Φ4,𝜑11𝑞=^∞_{𝑛=1Φ4=𝑛}𝑥²₁−9Φ142⁴𝑞^𝑛,Θ_Φ4,𝜑22𝑞=^∞_{𝑛=1Φ2=𝑛}𝑥²₂−1Φ71⁴𝑞^𝑛,Θ_{𝐹3⊕Φ1,𝜑11}𝑞=^∞_{𝑛=1𝐹3⊕Φ1=𝑛}𝑥²₁−9𝐹71³⊕Φ₁𝑞^𝑛,Θ_{𝐹3⊕Φ1,𝜑22}𝑞=^∞_{𝑛=1𝐹3⊕Φ1=𝑛}𝑥²₂−1𝐹142³⊕Φ₁𝑞^𝑛,Θ_{𝐹3⊕Φ1,𝜑77}𝑞=^∞_{𝑛=1𝐹3⊕Φ1=𝑛}𝑥²₇−9𝐹142³⊕Φ₁𝑞^𝑛,Θ_{𝐹3⊕Φ1,𝜑88}𝑞=^∞_{𝑛=1𝐹3⊕Φ1=𝑛}𝑥²₈−1𝐹71³⊕Φ₁𝑞^𝑛,Θ_{𝐹2⊕Φ2,𝜑11}𝑞=^∞_{𝑛=1Φ4=𝑛}𝑥²₁−9𝐹71²⊕Φ₂𝑞^𝑛,Θ_{𝐹2⊕Φ2,𝜑22}𝑞=^∞_{𝑛=1Φ4=𝑛}𝑥²₂−1𝐹142²⊕Φ₂𝑞^𝑛,Θ_{𝐹2⊕Φ2,𝜑55}𝑞=^∞_{𝑛=1Φ4=𝑛}𝑥²₅−9𝐹142²⊕Φ₂𝑞^𝑛,Θ_Ψ4,𝜑11𝑞=^∞_{𝑛=1Ψ4=𝑛}𝑥²₁−3Ψ71⁴𝑞^𝑛,Θ_Ψ4,𝜑12𝑞=^∞_{𝑛=1Ψ4=𝑛}𝑥₁𝑥₂−1Ψ284⁴𝑞^𝑛,Θ_Ψ4,𝜑22𝑞=^∞_{𝑛=1Ψ4=𝑛}𝑥²₂−3Ψ142⁴𝑞^𝑛,Θ_{𝐹3⊕Ψ1,𝜑11}𝑞=^∞_{𝑛=1𝐹3⊕Ψ1=𝑛}𝑥²₁−9𝐹71³⊕Ψ₁𝑞^𝑛,Θ_{𝐹2⊕Ψ2,𝜑22}𝑞=^∞_{𝑛=1𝐹2⊕Ψ2=𝑛}𝑥²₂−1𝐹142²⊕Ψ₂𝑞^𝑛,Θ_{𝐹2⊕Ψ2,𝜑66}𝑞=^∞_{𝑛=1𝐹2⊕Ψ2=𝑛}𝑥²₆−3𝐹142²⊕Ψ₂𝑞^𝑛,Θ_{Φ3⊕Ψ1,𝜑11}𝑞=^∞_{𝑛=1Φ3⊕Ψ1=𝑛}𝑥²₁−9Φ71³⊕Ψ₁𝑞^𝑛,Θ_{Φ3⊕Ψ1,𝜑22}𝑞=^∞_{𝑛=1Φ3⊕Ψ1=𝑛}𝑥²₂−1Φ71³⊕Ψ₁𝑞^𝑛,Θ_{Φ1⊕Ψ3,𝜑22}𝑞=^∞_{𝑛=1Φ1⊕Ψ3=𝑛}𝑥²₂−1Φ71¹⊕Ψ₃𝑞^𝑛,Θ_{Φ1⊕Ψ3,𝜑44}𝑞=^∞_{𝑛=1Φ1⊕Ψ3=𝑛}𝑥²₄−3Φ142¹⊕Ψ₃𝑞^𝑛,Θ_{Φ1⊕Ψ3,𝜑44}𝑞=^∞_{𝑛=1Φ1⊕Ψ3=𝑛}𝑥²₄−3Φ142¹⊕Ψ₃𝑞^𝑛,Θ_{𝐹3⊕Λ1,𝜑11}𝑞=^∞_{𝑛=1𝐹3⊕Λ1=𝑛}𝑥²₁−9𝐹71³⊕Λ₁𝑞^𝑛,Θ_{𝐹3⊕Λ1,𝜑22}𝑞=^∞_{𝑛=1𝐹3⊕Λ1=𝑛}𝑥²₂−1𝐹142³⊕Λ₁𝑞^𝑛,Θ_{Φ1⊕Λ3,𝜑12}𝑞=^∞_{𝑛=1Φ1⊕Λ3=𝑛}𝑥₁𝑥₂−1Φ284¹⊕Λ₃𝑞^𝑛,Θ_{Φ1⊕Λ3,𝜑22}𝑞=^∞_{𝑛=1Φ1⊕Λ3=𝑛}𝑥²₂−1Φ71¹⊕Λ₃𝑞^𝑛,Θ_{Φ1⊕Λ3,𝜑44}𝑞=^∞_{𝑛=1Φ1⊕Λ3=𝑛}𝑥²₄−2Φ71¹⊕Λ₃𝑞^𝑛,Θ_{Ψ3⊕Λ1,𝜑22}𝑞=^∞_{𝑛=1Ψ3⊕Λ1=𝑛}𝑥²₂−3Ψ142³⊕Λ₁𝑞^𝑛,Θ_{𝐹2⊕Φ1⊕Ψ1,𝜑55}𝑞=^∞_{𝑛=1𝐹2⊕Φ1⊕Ψ1=𝑛}𝑥²₅−9𝐹142²⊕Φ₁⊕Ψ₁𝑞^𝑛,Θ_{𝐹2⊕Φ1⊕Ψ1,𝜑56}𝑞=^∞_{𝑛=1𝐹2⊕Φ1⊕Ψ1=𝑛}𝑥₅𝑥₆−1𝐹284²⊕Φ₁⊕Ψ₁𝑞^𝑛,Θ_{𝐹2⊕Φ1⊕Ψ1,𝜑78}𝑞=^∞_{𝑛=1𝐹2⊕Φ1⊕Ψ1=𝑛}𝑥₇𝑥₈−1𝐹284²⊕Φ₁⊕Ψ₁𝑞^𝑛 (35) are a basis of 𝑆₆(Γ₀(71)).

Proof. 𝐹₂=𝑥²₁+𝑥₁𝑥₂+18𝑥²₂+𝑥²₃+𝑥₃𝑥₄+18𝑥²₄=𝑛 (36) has the following solutions for 𝑛=1⟹𝑥₁=±1,𝑥₂=0,𝑥₃=0,𝑥₄𝑥=0;₁=0,𝑥₂=0,𝑥₃=±1,𝑥₄=0,𝑛=2⟹𝑥₁=±1,𝑥₂=0,𝑥₃=±1,𝑥₄=0,𝑛=4⟹𝑥₁=±2,𝑥₂=0,𝑥₃=0,𝑥₄𝑥=0;₁=0,𝑥₂=0,𝑥₃=±2,𝑥₄=0,𝑛=5⟹𝑥₁=±1,𝑥₂=0,𝑥₃=±2,𝑥₄𝑥=0;₁=±2,𝑥₂=0,𝑥₃=±1,𝑥₄=0,𝑛=8⟹𝑥₁=±2,𝑥₂=0,𝑥₃=±2,𝑥₄=0,𝑛=9⟹𝑥₁=±3,𝑥₂=0,𝑥₃=0,𝑥₄𝑥=0;₁=0,𝑥₂=0,𝑥₃=±3,𝑥₄=0,𝑛=10⟹𝑥₁=±3,𝑥₂=0,𝑥₃=±1,𝑥₄𝑥=0;₁=±1,𝑥₂=0,𝑥₃=±3,𝑥₄=0,𝑛=13⟹𝑥₁=±2,𝑥₂=0,𝑥₃=±3,𝑥₄𝑥=0;₁=±3,𝑥₂=0,𝑥₃=±2,𝑥₄=0,𝑛=16⟹𝑥₁=±4,𝑥₂=0,𝑥₃=0,𝑥₄𝑥=0;₁=0,𝑥₂=0,𝑥₃=±4,𝑥₄=0,𝑛=17⟹𝑥₁=±1,𝑥₂=0,𝑥₃=±4,𝑥₄𝑥=0;₁=±4,𝑥₂=0,𝑥₃=±1,𝑥₄=0, (37) and there is no solutions for 𝑛=3,6,7,11,12,14,15. Therefore, Θ_𝐹2,𝜑11𝑞=171^∞_{𝑛=1𝐹2=𝑛}71𝑥²₁−9𝐹₂𝑞^𝑛=1)()𝑞71(71⋅2−9⋅4𝑞+71⋅4−9⋅2⋅4²+(71⋅4⋅2−9⋅4⋅4)𝑞⁴+(71⋅1⋅4+71⋅4⋅4−9⋅5⋅8)𝑞⁵+(71⋅4⋅4−9⋅8⋅4)𝑞⁸+(71⋅9⋅2−9⋅9⋅4)𝑞⁹+(71⋅9⋅4+71⋅1⋅4−9⋅10⋅8)𝑞¹⁰+(71⋅4⋅4+71⋅9⋅4−9⋅13⋅8)𝑞¹³+(71⋅16⋅2−9⋅16⋅4)𝑞¹⁶+(71⋅1⋅4+71⋅16⋅4−9⋅17⋅8)𝑞¹⁷=1+⋯71106𝑞+212𝑞²+424𝑞⁴+1060𝑞⁵+848𝑞⁸+954𝑞⁹+2120𝑞¹⁰+2756𝑞¹³+1696𝑞¹⁶+3604𝑞¹⁷+⋯. (38) Similarly, we have found the remaining 16 linearly independent theta series as follows: Θ_Φ2,𝜑11𝑞=12⋅71^∞_{𝑛=1Φ2=𝑛}2⋅71𝑥²₁−9Φ₂𝑞^𝑛=17170𝑞²+140𝑞⁴+280𝑞⁸−324𝑞⁹+482𝑞¹⁰−508𝑞¹¹−586𝑞¹²−440𝑞¹⁴+28𝑞¹⁵+560𝑞¹⁶+108𝑞¹⁷,Θ+⋯_Φ2,𝜑12𝑞=1142^∞_{𝑛=1Φ2=𝑛}142𝑥₁𝑥₂−Φ₂𝑞^𝑛=171−4𝑞²−8𝑞⁴−16𝑞⁸−18𝑞⁹+82𝑞¹⁰−44𝑞¹¹+70𝑞¹²−340𝑞¹⁴+254𝑞¹⁵−32𝑞¹⁶+432𝑞¹⁷,Θ+⋯_{𝐹1⊕Φ1,𝜑11}𝑞=171^∞_{𝑛=1Φ2=𝑛}71𝑥²₁−9𝐹₁⊕Φ₁𝑞^𝑛=171106𝑞−72𝑞²+68𝑞³+424𝑞⁴+704𝑞⁶−288𝑞⁸+266𝑞⁹−796𝑞¹⁰+1256𝑞¹¹−160𝑞¹²−452𝑞¹³+128𝑞¹⁴−540𝑞¹⁵+812𝑞¹⁶+1332𝑞¹⁷,Θ+⋯_{𝐹1⊕Φ1,𝜑12}𝑞=1142^∞_{𝑛=1Φ2=𝑛}142𝑥₁𝑥₂+𝐹₁⊕Φ₁𝑞^𝑛=171𝑞+2𝑞²+6𝑞³+4𝑞⁴+12𝑞⁶+8𝑞⁸+36𝑞⁹+30𝑞¹⁰+44𝑞¹¹+36𝑞¹²+52𝑞¹³+28𝑞¹⁴+15𝑞¹⁵+80𝑞¹⁶+34𝑞¹⁷,Θ+⋯_{𝐹1⊕Φ1,𝜑33}𝑞=171^∞_{𝑛=1Φ2=𝑛}71𝑥²₃−9𝐹₁⊕Φ₁𝑞^𝑛=171−18𝑞+106𝑞²+176𝑞³−72𝑞⁴+68𝑞⁶+424𝑞⁸+488𝑞⁹−398𝑞¹⁰−224𝑞¹¹+630𝑞¹²−652𝑞¹³−220𝑞¹⁴+298𝑞¹⁵−20𝑞¹⁶+524𝑞¹⁷,Θ+⋯_{𝐹1⊕Φ1,𝜑34}𝑞=1142^∞_{𝑛=1𝐹1⊕Φ1=𝑛}142𝑥₃𝑥₄−𝐹₁⊕Φ₁𝑞^𝑛=171−𝑞−2𝑞²−6𝑞³−4𝑞⁴−12𝑞⁶−8𝑞⁸−36𝑞⁹+112𝑞¹⁰+240𝑞¹¹−178𝑞¹²−336𝑞¹³+256𝑞¹⁴+269𝑞¹⁵+204𝑞¹⁶−34𝑞¹⁷,Θ+⋯_Ψ2,𝜑11𝑞=171^∞_{𝑛=1Ψ2=𝑛}71𝑥²₁−6Ψ₂𝑞^𝑛=17170𝑞³−4𝑞⁶−50𝑞⁸−148𝑞⁹−98𝑞¹⁰+40𝑞¹¹−8𝑞¹²−56𝑞¹³−388𝑞¹⁴+700𝑞¹⁵−400𝑞¹⁶,Θ+⋯_{𝐹1⊕Ψ1,𝜑11}𝑞=171^∞_{𝑛=1𝐹1⊕Ψ1=𝑛}71𝑥²₁−18𝐹₁⊕Ψ₁𝑞^𝑛=171106𝑞−108𝑞³+420𝑞⁴−216𝑞⁶+412𝑞⁷−288𝑞⁸+590𝑞⁹+56𝑞¹⁰−508𝑞¹¹+1532𝑞¹²−652𝑞¹³+128𝑞¹⁴+1476𝑞¹⁵+1104𝑞¹⁶+392𝑞¹⁷,Θ+⋯_{𝐹1⊕Ψ1,𝜑12}𝑞=1142^∞_{𝑛=1𝐹1⊕Ψ1=𝑛}142𝑥₁𝑥₂+𝐹₁⊕Ψ₁𝑞^𝑛=171𝑞+3𝑞³+12𝑞⁴+6𝑞⁶+28𝑞⁷+8𝑞⁸+27𝑞⁹+30𝑞¹⁰+22𝑞¹¹+60𝑞¹²+26𝑞¹³+28𝑞¹⁴+30𝑞¹⁵+64𝑞¹⁶+68𝑞¹⁷,Θ+⋯_{𝐹1⊕Ψ1,𝜑33}𝑞=171^∞_{𝑛=1𝐹1⊕Ψ1=𝑛}71𝑥₁𝑥₂−6𝐹₁⊕Ψ₁𝑞^𝑛=171−12𝑞+106𝑞³+140𝑞⁴−72𝑞⁶−52𝑞⁷+46𝑞⁸−40𝑞⁹−218𝑞¹⁰+20𝑞¹¹+416𝑞¹²+824𝑞¹³−52𝑞¹⁴−360𝑞¹⁵+936𝑞¹⁶+604𝑞¹⁷,Θ+⋯_{𝐹1⊕Ψ1,𝜑44}𝑞=171^∞_{𝑛=1𝐹1⊕Ψ1=𝑛}71𝑥²₄−3𝐹₁⊕Ψ₁𝑞^𝑛=171−6𝑞−18𝑞³−72𝑞⁴+106𝑞⁶+116𝑞⁷+94𝑞⁸+122𝑞⁹+246𝑞¹⁰+152𝑞¹¹−76𝑞¹²−156𝑞¹³+116𝑞¹⁴+104𝑞¹⁵−242𝑞¹⁶+160𝑞¹⁷,Θ+⋯_{Φ1⊕Ψ1,𝜑11}𝑞=171^∞_{𝑛=1Ψ2=𝑛}71𝑥²₁−9Φ₁⊕Ψ₁𝑞^𝑛=171106𝑞²−54𝑞³+104𝑞⁵−108𝑞⁶+276𝑞⁸−162𝑞⁹−294𝑞¹⁰+740𝑞¹¹−870𝑞¹²−184𝑞¹³+412𝑞¹⁴−498𝑞¹⁵−20𝑞¹⁶−612𝑞¹⁷,Θ+⋯_{Φ1⊕Ψ1,𝜑34}𝑞=1142^∞_{𝑛=1Ψ2=𝑛}142𝑥₃𝑥₄−Φ₁⊕Ψ₁𝑞^𝑛=171−2𝑞²−3𝑞³−10𝑞⁵−6𝑞⁶+110𝑞⁸−9𝑞⁹+102𝑞¹⁰−22𝑞¹¹−356𝑞¹²−26𝑞¹³−56𝑞¹⁴−75𝑞¹⁵+488𝑞¹⁶+250𝑞¹⁷,Θ+⋯_Λ2,𝜑11𝑞=171^∞_{𝑛=1Λ2=𝑛}71𝑥²₂−4Λ₂𝑞^𝑛=171−64𝑞⁴+62𝑞⁵+46𝑞⁶−128𝑞⁸−4𝑞⁹+88𝑞¹⁰+216𝑞¹¹+42𝑞¹²−98𝑞¹⁵−484𝑞¹⁶+24𝑞¹⁷,Θ+⋯_{𝐹1⊕Λ1,𝜑11}𝑞=171^∞_{𝑛=1𝐹1⊕Λ1=𝑛}71𝑥²₁−18𝐹₁⊕Λ₁𝑞^𝑛=171134𝑞+504𝑞⁴+164𝑞⁵+140𝑞⁶+172𝑞⁷+1008𝑞⁸+2198𝑞⁹+976𝑞¹⁰−96𝑞¹²+2424𝑞¹³+2332𝑞¹⁴+2196𝑞¹⁵+2924𝑞¹⁶+12𝑞¹⁷,Θ+⋯_{Ψ1⊕Λ1,𝜑34}𝑞=1142^∞_{𝑛=1Ψ1⊕Λ1=𝑛}142𝑥²₁−3Ψ₁⊕Λ₁𝑞^𝑛=171−24𝑞³−32𝑞⁴−40𝑞⁵+46𝑞⁶−112𝑞⁷−192𝑞⁸+140𝑞⁹−240𝑞¹⁰−176𝑞¹¹−434𝑞¹²−208𝑞¹³−164𝑞¹⁴−600𝑞¹⁵−484𝑞¹⁶−272𝑞¹⁷+⋯. (39)
Since the 17th determinant of the coefficients of the theta series is 22370813204653670400000,9095120158391 (40) the first 17 generalized theta series in the theorem are a basis of 𝑆₄(Γ₀(71)). Similarly, the second 29 generalized theta series in the theorem are the following, and they are a basis of 𝑆₆(Γ₀(71)) since the 29th determinant of the coefficients is−6041160948656511821438109137218074776490167579085845299200.231122292121701565271 (41)
Therefore, Θ_𝐹4,𝜑11𝑞=17170𝑞+420𝑞²+840𝑞³+840𝑞⁴+2100𝑞⁵+5040𝑞⁶+3920𝑞⁷+1680𝑞⁸+8190𝑞⁹+12600𝑞¹⁰+9240𝑞¹¹+10080𝑞¹²+12740𝑞¹³+23520𝑞¹⁴+25200𝑞¹⁵+3360𝑞¹⁶+21420𝑞¹⁷+46690𝑞¹⁸+12740𝑞¹⁹−3010𝑞²⁰+18620𝑞²¹+21420𝑞²²−17360𝑞²³−29750𝑞²⁴+10150𝑞²⁵+12600𝑞²⁶+30380𝑞²⁷+2940𝑞²⁸−10080𝑞²⁹,Θ+⋯_Φ4,𝜑11𝑞=17170𝑞²+420𝑞⁴+840𝑞⁶+840𝑞⁸−324𝑞⁹+1882𝑞¹⁰−1524𝑞¹¹+3862𝑞¹²−2208𝑞¹³+1664𝑞¹⁴−884𝑞¹⁵−184𝑞¹⁶+324𝑞¹⁷+5978𝑞¹⁸−728𝑞¹⁹+7464𝑞²⁰−11080𝑞²¹+1992𝑞²²−20568𝑞²³−3966𝑞²⁴+140𝑞²⁵−5044𝑞²⁶+6288𝑞²⁷+14688𝑞²⁸−26740𝑞²⁹,Θ+⋯_Φ4,𝜑22𝑞=171−16𝑞²−96𝑞⁴−192𝑞⁶−192𝑞⁸+70𝑞⁹−418𝑞¹⁰+324𝑞¹¹−830𝑞¹²+456𝑞¹³−356𝑞¹⁴+198𝑞¹⁵−104𝑞¹⁶+72𝑞¹⁷−1480𝑞¹⁸+422𝑞¹⁹−1560𝑞²⁰+2492𝑞²¹−504𝑞²²+4044𝑞²³+586𝑞²⁴−32𝑞²⁵+1372𝑞²⁶+72𝑞²⁷−2700𝑞²⁸+6112𝑞²⁹,Θ+⋯_{𝐹3⊕Φ1,𝜑11}𝑞=17188𝑞+316𝑞²+312𝑞³+624𝑞⁴+2176𝑞⁵+2600𝑞⁶+2656𝑞⁷+4624𝑞⁸+3110𝑞⁹+3904𝑞¹⁰+8120𝑞¹¹+5160𝑞¹²+9396𝑞¹³+10520𝑞¹⁴+6530𝑞¹⁵+18348𝑞¹⁶+12240𝑞¹⁷+14862𝑞¹⁸+29326𝑞¹⁹+13910𝑞²⁰+12000𝑞²¹+13540𝑞²²+21844𝑞²³+3578𝑞²⁴+2948𝑞²⁵+62160𝑞²⁶+21596𝑞²⁷+24068𝑞²⁸+44156𝑞²⁹,Θ+⋯_{𝐹3⊕Φ1,𝜑22}𝑞=171−3𝑞−14𝑞²−30𝑞³−60𝑞⁴−100𝑞⁵−108𝑞⁶−168𝑞⁷−248𝑞⁸−198𝑞⁹−430𝑞¹⁰−748𝑞¹¹−660𝑞¹²−1040𝑞¹³−1372𝑞¹⁴−1215𝑞¹⁵−1584𝑞¹⁶−1734𝑞¹⁷−2038𝑞¹⁸−2189𝑞¹⁹−2012𝑞²⁰−1842𝑞²¹−652𝑞²²−1942𝑞²³−1308𝑞²⁴+77𝑞²⁵−3836𝑞²⁶−1088𝑞²⁷−856𝑞²⁸−4568𝑞²⁹,Θ+⋯_{𝐹3⊕Φ1,𝜑77}𝑞=171−27𝑞+16𝑞²+582𝑞³+1164𝑞⁴+236𝑞⁵−120𝑞⁶+1896𝑞⁷+1744𝑞⁸+1626𝑞⁹+4792𝑞¹⁰+2924𝑞¹¹+2722𝑞¹²+9668𝑞¹³+4976𝑞¹⁴−2415𝑞¹⁵+7044𝑞¹⁶+11658𝑞¹⁷+4520𝑞¹⁸+6427𝑞¹⁹+12848𝑞²⁰+14946𝑞²¹+4072𝑞²²+2970𝑞²³+25574𝑞²⁴+8361𝑞²⁵+5520𝑞²⁶+25424𝑞²⁷+7632𝑞²⁸+26480𝑞²⁹⋯,Θ_{𝐹3⊕Φ1,𝜑88}𝑞=171−6𝑞−28𝑞²−60𝑞³−120𝑞⁴−200𝑞⁵−216𝑞⁶−336𝑞⁷−496𝑞⁸−254𝑞⁹+134𝑞¹⁰+1060𝑞¹¹+1662𝑞¹²+760𝑞¹³+3220𝑞¹⁴+5664𝑞¹⁵+1944𝑞¹⁶+3348𝑞¹⁷+5864𝑞¹⁸+2012𝑞¹⁹+4780𝑞²⁰+3416𝑞²¹−2440𝑞²²+3216𝑞²³+1218𝑞²⁴−3822𝑞²⁵+564𝑞²⁶−6152𝑞²⁷−4836𝑞²⁸+7336𝑞²⁹⋯,Θ_{𝐹2⊕Φ2𝐹4,𝜑11}𝑞=171106𝑞+140𝑞²+136𝑞³+840𝑞⁴+908𝑞⁵+1680𝑞⁶+3664𝑞⁷+1680𝑞⁸+2990𝑞⁹+2968𝑞¹⁰−760𝑞¹¹+5872𝑞¹²+2948𝑞¹³+6096𝑞¹⁴+11028𝑞¹⁵+1336𝑞¹⁶+14948𝑞¹⁷+9018q¹⁸+11416𝑞¹⁹+19926𝑞²⁰+4804𝑞²¹+11468𝑞²²−8616𝑞²³+17986𝑞²⁴+20146𝑞²⁵+9704𝑞²⁶+69068𝑞²⁷+29308𝑞²⁸+5752𝑞²⁹⋯,Θ_{𝐹2⊕Φ2,𝜑22}𝑞=171−2𝑞−8𝑞²−24𝑞³−48𝑞⁴−60𝑞⁵−96𝑞⁶−112𝑞⁷−96𝑞⁸−252𝑞⁹−340𝑞¹⁰−484𝑞¹¹−936𝑞¹²−988𝑞¹³−1176𝑞¹⁴−1470𝑞¹⁵−1472𝑞¹⁶−1836𝑞¹⁷−1984𝑞¹⁸−3042𝑞¹⁹−2940𝑞²⁰−2368𝑞²¹−4112𝑞²²−1804𝑞²³−2748𝑞²⁴−5610𝑞²⁵−2648𝑞²⁶−5740𝑞²⁷−4888𝑞²⁸−1416𝑞²⁹⋯,Θ_{𝐹2⊕Φ2,𝜑55}𝑞=171−18𝑞+70𝑞²+352𝑞³+420𝑞⁴+596𝑞⁵+840𝑞⁶+128𝑞⁷+840𝑞⁸+2276𝑞⁹+1342𝑞¹⁰+2744𝑞¹¹+4214𝑞¹²+764𝑞¹³+1344𝑞¹⁴+4378𝑞¹⁵−184𝑞¹⁶+3072𝑞¹⁷+10686𝑞¹⁸+4714𝑞¹⁹+7904𝑞²⁰+9928𝑞²¹+7864𝑞²²+3076𝑞²³−1302𝑞²⁴+22498𝑞²⁵+4852𝑞²⁶+9684𝑞²⁷+31552𝑞²⁸−7348𝑞²⁹⋯,Θ_Ψ4,𝜑11𝑞=171130𝑞+236𝑞²+424𝑞³+1416𝑞⁴+1628𝑞⁵+2832𝑞⁶+5008𝑞⁷+2832𝑞⁸+6014𝑞⁹+7048𝑞¹⁰+5048𝑞¹¹+17104𝑞¹²+14804𝑞¹³+20208𝑞¹⁴+28668𝑞¹⁵+19000𝑞¹⁶+36980𝑞¹⁷+36234𝑞¹⁸+54736𝑞¹⁹+72246𝑞²⁰+67300𝑞²¹+94892𝑞²²+67560𝑞²³+102082𝑞²⁴+118296𝑞²⁵+116512𝑞²⁶+200020𝑞²⁷+190620𝑞²⁸+213692𝑞²⁹⋯,Θ_Ψ4,𝜑12𝑞=171−𝑞−4𝑞²−12𝑞³−24𝑞⁴−30𝑞⁵−48𝑞⁶−56𝑞⁷−48𝑞⁸−126𝑞⁹−170𝑞¹⁰−242𝑞¹¹−468𝑞¹²−494𝑞¹³−588𝑞¹⁴−735𝑞¹⁵−736𝑞¹⁶−918𝑞¹⁷−1276𝑞¹⁸−2089𝑞¹⁹−2890𝑞²⁰−4024𝑞²¹−4896𝑞²²−5446𝑞²³−5634𝑞²⁴−5620𝑞²⁵−8088𝑞²⁶−9010𝑞²⁷−13012𝑞²⁸−18142𝑞²⁹,Θ+⋯_Ψ4,𝜑22𝑞=171−6𝑞−24𝑞²−72𝑞³−144𝑞⁴−180𝑞⁵−288𝑞⁶−336𝑞⁷−288𝑞⁸−756𝑞⁹−1020𝑞¹⁰−1452𝑞¹¹−2808𝑞¹²−2964𝑞¹³−3528𝑞¹⁴−4410𝑞¹⁵−4416𝑞¹⁶−5508𝑞¹⁷−6520𝑞¹⁸−10262𝑞¹⁹−11660𝑞²⁰−12784𝑞²¹−18016𝑞²²−14500𝑞²³−16764𝑞²⁴−22360𝑞²⁵−21264𝑞²⁶−29068𝑞²⁷−34904𝑞²⁸−38420𝑞²⁹,Θ+⋯_{𝐹3⊕Ψ1,𝜑11}𝑞=17188𝑞+352𝑞²+298𝑞³+204𝑞⁴+1816𝑞⁵+2276𝑞⁶−92𝑞⁷+2896𝑞⁸+6040𝑞⁹+2372𝑞¹⁰+5616𝑞¹¹+7508𝑞¹²+8528𝑞¹³+16328𝑞¹⁴+5240𝑞¹⁵+13824𝑞¹⁶+34624𝑞¹⁷+10182𝑞¹⁸+14444𝑞¹⁹+38886𝑞²⁰+27684𝑞²¹+7336𝑞²²+11636𝑞²³+40714𝑞²⁴+478𝑞²⁵+3156𝑞²⁶+23902𝑞²⁷−29536𝑞²⁸+38042𝑞²⁹,Θ+⋯_{𝐹2⊕Ψ2,𝜑22}𝑞=171−2𝑞−4𝑞²−6𝑞³−40𝑞⁴−60𝑞⁵−24𝑞⁶−168𝑞⁷−288𝑞⁸−126𝑞⁹−460𝑞¹⁰−748𝑞¹¹−528𝑞¹²−1144𝑞¹³−1288𝑞¹⁴−1260𝑞¹⁵−2336𝑞¹⁶−2108𝑞¹⁷−2128𝑞¹⁸−3004𝑞¹⁹−3556𝑞²⁰−3000𝑞²¹−2088𝑞²²−4844𝑞²³−2572𝑞²⁴−1416𝑞²⁵−5432𝑞²⁶+1238𝑞²⁷−1816𝑞²⁸−5598𝑞²⁹,Θ+⋯_{𝐹2⊕Ψ2,𝜑66}𝑞=171−6𝑞−12𝑞²−18𝑞³−120𝑞⁴−180𝑞⁵+70𝑞⁶+64𝑞⁷−154𝑞⁸+474𝑞⁹+1034𝑞¹⁰+880𝑞¹¹+972𝑞¹²+1396𝑞¹³+2384𝑞¹⁴+2468𝑞¹⁵+1370𝑞¹⁶+2764𝑞¹⁷+4976𝑞¹⁸+2632𝑞¹⁹+1118𝑞²⁰+3496𝑞²¹+836𝑞²²+520𝑞²³−48𝑞²⁴−3680𝑞²⁵+5288𝑞²⁶−262𝑞²⁷−5164𝑞²⁸+7630𝑞²⁹,Θ+⋯_{Φ3⊕Ψ1,𝜑11}𝑞=171106𝑞+212𝑞²−108𝑞³+416𝑞⁴+1476𝑞⁵−432𝑞⁶+384𝑞⁷+3904𝑞⁸−422𝑞⁹+1376𝑞¹⁰+5280𝑞¹¹+152𝑞¹²+8092𝑞¹³+2944𝑞¹⁴+1176𝑞¹⁵+21568𝑞¹⁶+1532𝑞¹⁷+5858𝑞¹⁸+30276𝑞¹⁹+13382𝑞²⁰+12172𝑞²¹+20636𝑞²²+42880𝑞²³+7522𝑞²⁴+11006𝑞²⁵+65240𝑞²⁶−17192𝑞²⁷+27804𝑞²⁸+44644𝑞²⁹⋯,Θ_{Φ3⊕Ψ1,𝜑22}𝑞=171−4𝑞−8𝑞²−12𝑞³−80𝑞⁴−120𝑞⁵−48𝑞⁶−336𝑞⁷−576𝑞⁸−252𝑞⁹−920𝑞¹⁰−1496𝑞¹¹−1056𝑞¹²−2288𝑞¹³−2576𝑞¹⁴−2520𝑞¹⁵−4672𝑞¹⁶−4216𝑞¹⁷−4540𝑞¹⁸−6576𝑞¹⁹−7396𝑞²⁰−7704𝑞²¹−7016𝑞²²−10824𝑞²³−10540𝑞²⁴−10216𝑞²⁵−14272𝑞²⁶−10588𝑞²⁷−17832𝑞²⁸−20852𝑞²⁹⋯,Θ_{Φ1⊕Ψ3,𝜑22}𝑞=171−4𝑞²−18𝑞³−60𝑞⁵−108𝑞⁶−352𝑞⁸−164𝑞⁹−58𝑞¹⁰−1100𝑞¹¹+178𝑞¹²−240𝑞¹³−2352𝑞¹⁴+1360𝑞¹⁵−804𝑞¹⁶−2888𝑞¹⁷+2504𝑞¹⁸−504𝑞¹⁹−2528𝑞²⁰+4292𝑞²¹−12𝑞²²+960𝑞²³+3354𝑞²⁴+5154𝑞²⁵+1556𝑞²⁶+5066𝑞²⁷+6812𝑞²⁸+6342𝑞²⁹,Θ+⋯_{Φ1⊕Ψ3,𝜑44}𝑞=171−6𝑞²−27𝑞³−90𝑞⁵−20𝑞⁶−102𝑞⁸+109𝑞⁹+126𝑞¹⁰+54𝑞¹¹+196𝑞¹²+66𝑞¹³+732𝑞¹⁴−729𝑞¹⁵+1350𝑞¹⁶+354𝑞¹⁷−362𝑞¹⁸+2013𝑞¹⁹+42𝑞²⁰−946𝑞²¹+2112𝑞²²−2394𝑞²³+984𝑞²⁴−3771𝑞²⁵+3612𝑞²⁶−5891𝑞²⁷+420𝑞²⁸−2415𝑞²⁹⋯,Θ_{𝐹3⊕Λ1,𝜑11}𝑞=17188𝑞+352𝑞²+352𝑞³+280𝑞⁴+1414𝑞⁵+1480𝑞⁶−720𝑞⁷+1072𝑞⁸+5756𝑞⁹+7432𝑞¹⁰+6864𝑞¹¹+4776𝑞¹²+10952𝑞¹³+20020𝑞¹⁴+14282𝑞¹⁵+11356𝑞¹⁶+20888𝑞¹⁷+27410𝑞¹⁸+17852𝑞¹⁹+9398𝑞²⁰+24364𝑞²¹+27004𝑞²²+13348𝑞²³−2814𝑞²⁴−2492𝑞²⁵+15596𝑞²⁶−17286𝑞²⁷−37880𝑞²⁸+12452𝑞²⁹⋯,Θ_{𝐹3⊕Λ1,𝜑22}𝑞=171−3𝑞−12𝑞²−12𝑞³−16𝑞⁴−95𝑞⁵−186𝑞⁶−182𝑞⁷−256𝑞⁸−477𝑞⁹−660𝑞¹⁰−660𝑞¹¹−492𝑞¹²−780𝑞¹³−1428𝑞¹⁴−1305𝑞¹⁵−1152𝑞¹⁶−2074𝑞¹⁷−2398𝑞¹⁸−1296𝑞¹⁹−1600𝑞²⁰−2938𝑞²¹−2176𝑞²²−1420𝑞²³−748𝑞²⁴−383𝑞²⁵−1800𝑞²⁶+1075𝑞²⁷+2976𝑞²⁸−1880𝑞²⁹,Θ+⋯_{Φ1⊕Λ3,𝜑12}𝑞=1142−2𝑞²−12𝑞⁴−15𝑞⁵−54𝑞⁶−42𝑞⁷−104𝑞⁸−117𝑞⁹−26𝑞¹⁰−396𝑞¹¹−956𝑞¹²−624𝑞¹³+752𝑞¹⁴+1252𝑞¹⁵−1776𝑞¹⁶−3880𝑞¹⁷−1464𝑞¹⁸+6787𝑞¹⁹+6604𝑞²⁰−288𝑞²¹−8956𝑞²²−4090𝑞²³+6476𝑞²⁴+17282𝑞²⁵+9308𝑞²⁶−8255𝑞²⁷−22264𝑞²⁸−11766𝑞²⁹,Θ+⋯_{Φ1⊕Λ3,𝜑22}𝑞=171−4𝑞²−24𝑞⁴−30𝑞⁵−108𝑞⁶−84𝑞⁷−208𝑞⁸−92𝑞⁹−478𝑞¹⁰−792𝑞¹¹−1202𝑞¹²−396𝑞¹³−200𝑞¹⁴−194𝑞¹⁵−1848𝑞¹⁶−1796𝑞¹⁷−372𝑞¹⁸+3776𝑞¹⁹+3836𝑞²⁰+3968𝑞²¹−20𝑞²²+1192𝑞²³+2018𝑞²⁴+8152𝑞²⁵+2428𝑞²⁶+1950𝑞²⁷−4484𝑞²⁸+324𝑞²⁹,Θ+⋯_{Φ1⊕Λ3,𝜑44}𝑞=171−8𝑞²−48𝑞⁴+82𝑞⁵−74𝑞⁶+116𝑞⁷−132𝑞⁸+100𝑞⁹−104𝑞¹⁰+688𝑞¹¹+294𝑞¹²+628𝑞¹³−116𝑞¹⁴+606𝑞¹⁵−4𝑞¹⁶+1520𝑞¹⁷+108𝑞¹⁸+452𝑞¹⁹−1416𝑞²⁰−3424𝑞²¹−4868𝑞²²−2160𝑞²³−224𝑞²⁴+1536𝑞²⁵+2584𝑞²⁶−2206𝑞²⁷−4424𝑞²⁸−1056𝑞²⁹,Θ+⋯_{Ψ3⊕Λ1,𝜑22}𝑞=171−27𝑞³−12𝑞⁴−15𝑞⁵−38𝑞⁶−126𝑞⁷−74𝑞⁸−26𝑞⁹−204𝑞¹⁰−138𝑞¹¹+48𝑞¹²+38𝑞¹³+552𝑞¹⁴−493𝑞¹⁵+1018𝑞¹⁶+1002𝑞¹⁷−294𝑞¹⁸+874𝑞¹⁹+1654𝑞²⁰−286𝑞²¹+1816𝑞²²+48𝑞²³+308𝑞²⁴−561𝑞²⁵+1552𝑞²⁶−6580𝑞²⁷+1716𝑞²⁸−3853𝑞²⁹,Θ+⋯_{𝐹2⊕Φ1⊕Λ1,𝜑55}𝑞=171−18𝑞+88𝑞²+433𝑞³+208𝑞⁴−166𝑞⁵+1002𝑞⁶+760𝑞⁷−444𝑞⁸+2357𝑞⁹+1994𝑞¹⁰−354𝑞¹¹+4954𝑞¹²+3954𝑞¹³−1972𝑞¹⁴+5933𝑞¹⁵+8148𝑞¹⁶−1626𝑞¹⁷+5256𝑞¹⁸+11771𝑞¹⁹+6740𝑞²⁰+5090𝑞²¹+4468𝑞²²+18450𝑞²³+5798𝑞²⁴+3171𝑞²⁵+27768𝑞²⁶−3535𝑞²⁷+20296𝑞²⁸+21929𝑞²⁹,Θ+⋯_{𝐹2⊕Φ1⊕Λ1,𝜑56}𝑞=1142−2𝑞−6𝑞²−15𝑞³−40𝑞⁴−50𝑞⁵−78𝑞⁶−168𝑞⁷−144𝑞⁸−243𝑞⁹−236𝑞¹⁰+718𝑞¹¹+156𝑞¹²−1738𝑞¹³+1264𝑞¹⁴+3089𝑞¹⁵−2408𝑞¹⁶+198𝑞¹⁷+6406𝑞¹⁸+1939𝑞¹⁹+1380𝑞²⁰−34𝑞²¹+2232𝑞²²+7730𝑞²³−8996𝑞²⁴+1299𝑞²⁵+10564𝑞²⁶−8471𝑞²⁷+13552𝑞²⁸−2139𝑞²⁹,Θ+⋯_{𝐹2⊕Φ1⊕Λ1,𝜑78}𝑞=171−2𝑞−6𝑞²−15𝑞³−40𝑞⁴−50𝑞⁵−78𝑞⁶−168𝑞⁷+140𝑞⁸+893𝑞⁹+900𝑞¹⁰+718𝑞¹¹+1008𝑞¹²−1170𝑞¹³−2144𝑞¹⁴+817𝑞¹⁵−1840𝑞¹⁶−370𝑞¹⁷+8678𝑞¹⁸+4211𝑞¹⁹+1948𝑞²⁰+2238𝑞²¹−5152𝑞²²−4198𝑞²³−11552𝑞²⁴−923𝑞²⁵+14076𝑞²⁶+2997𝑞²⁷+18432𝑞²⁸+6161𝑞²⁹+⋯. (42)

5. Representation Numbers of 𝑁

Proposition 5. The differences between the following theta series of the quadratic forms (4) and the Eisenstein series (22) are the linear combinations of the generalized theta series in the preceding theorem. The coefficients are given in the table [12]. Similarly, the differences between the following theta series of the quadratic forms (5) and the Eisenstein series (23) are the linear combinations of the generalized theta series in the preceding theorem. The coefficients are given in the table [12].

Proof. Let us see the situation in the following case: Θ_𝐹4𝑞−𝐸_𝐹4𝑞=200482521𝑞+59424𝑞2521²+77312𝑞2521³+51744𝑞2521⁴+105888𝑞2521⁵+211776𝑞2521⁶+120064𝑞2521⁷−9696𝑞2521⁸+171344𝑞2521⁹+226944𝑞2521¹⁰+82176𝑞2521¹¹−3264𝑞2521¹²+18592𝑞2521¹³+112512𝑞2521¹⁴+60672𝑞2521¹⁵−501216𝑞2521¹⁶−226656𝑞2521¹⁷+⋯=𝑐₁Θ_𝐹2,𝜑11𝑞+𝑐₂Θ_Φ2,𝜑11𝑞+𝑐₃Θ_Φ2,𝜑12𝑞+𝑐₄Θ_{𝐹1⊕Φ1,𝜑11}𝑞+𝑐₅Θ_{𝐹1⊕Φ1,𝜑12}𝑞+𝑐₆Θ_{𝐹1⊕Φ1,𝜑33}𝑞+𝑐₇Θ_{𝐹1⊕Φ1,𝜑34}𝑞+𝑐₈Θ_Ψ2,𝜑11𝑞+𝑐₉Θ_{𝐹1⊕Ψ1,𝜑11}𝑞+𝑐₁₀Θ_{𝐹1⊕Ψ1,𝜑12}𝑞+𝑐₁₁Θ_{𝐹1⊕Ψ1,𝜑33}𝑞+𝑐₁₂Θ_{𝐹1⊕Ψ1,𝜑44}𝑞+𝑐₁₃Θ_{Φ1⊕Ψ1,𝜑11}𝑞+𝑐₁₄Θ_{Φ1⊕Ψ1,𝜑34}𝑞+𝑐₁₅Θ_Λ2,𝜑22𝑞+𝑐₁₆Θ_{𝐹1⊕Λ1,𝜑11}𝑞+𝑐₁₇Θ_{Φ1⊕Λ1,𝜑34}𝑞. (43) By equating the coefficients of 𝑞^𝑛 in both sides for 𝑛=1,2,3,…,17, we get an equation in coefficients. We repeat the same procedure for the other cases. At the end, by solving 17 linear equations in 17 variables, we get the coefficients in table [12]. Here, the coefficients 𝑐₁₆ and 𝑐₁₇ are always zero. Similarly, we can do for the difference between the theta series of the quadratic forms (5) and the Eisenstein series (23).

Corollary 6. The representation numbers for the quadratic forms (4) (in these direct sums any form can be replaced by its inverse) are 𝑟()=𝑛;𝐻120𝜎2521^∗3(𝑛)+𝑐₁^∞_{𝑛=1𝐹2=𝑛}𝑥²₁−18𝐹71²𝑞^𝑛+𝑐_2Φ2=𝑛𝑥²₁−9𝑛71+𝑐_3Φ2=𝑛𝑥₁𝑥₂−1𝑛142+𝑐_{4𝐹1⊕Φ1=𝑛}𝑥²₁−18𝑛71+𝑐_{5𝐹1⊕Φ1=𝑛}𝑥₁𝑥₂+1𝑛142+𝑐_{6𝐹1⊕Φ1=𝑛}𝑥²₃−9𝑛71+𝑐_{7𝐹1⊕Φ1=𝑛}𝑥₃𝑥₄−1𝑛142+𝑐_8Ψ2=𝑛𝑥²₁−6𝑛71+𝑐_{9𝐹1⊕Ψ1=𝑛}𝑥²₁−18𝑛71+𝑐_{10𝐹1⊕Ψ1=𝑛}𝑥₁𝑥₂+1𝑛142+𝑐_{11𝐹1⊕Ψ1=𝑛}𝑥²₃−6𝑛71+𝑐_{12𝐹1⊕Ψ1=𝑛}𝑥²₃−3𝑛71+𝑐_{13Φ1⊕Ψ1=𝑛}𝑥²₁−9𝑛71+𝑐_{14Φ1⊕Ψ1=𝑛}𝑥₃𝑥₄−1𝑛142+𝑐_15Λ2=𝑛𝑥²₂−4𝑛.71 (44)
Similarly, the representation numbers for the quadratic forms (5) are 𝑟()=𝑛;𝐻36𝜎25565^∗5(𝑛)+𝑑₁^∞_{𝑛=1𝐹4=𝑛}𝑥²₁−9𝐹71⁴𝑞^𝑛+𝑑₂^∞_{𝑛=1Φ4=𝑛}𝑥²₁−9Φ142⁴𝑞^𝑛+𝑑₃^∞_{𝑛=1Φ2=𝑛}𝑥²₂−1Φ71⁴𝑞^𝑛+𝑑₄^∞_{𝑛=1𝐹3⊕Φ1=𝑛}𝑥²₁−9𝐹71³⊕Φ₁𝑞^𝑛+𝑑₅^∞_{𝑛=1𝐹3⊕Φ1=𝑛}𝑥²₂−1𝐹142³⊕Φ₁𝑞^𝑛+𝑑₆^∞_{𝑛=1𝐹3⊕Φ1=𝑛}𝑥²₇−9𝐹142³⊕Φ₁𝑞^𝑛+𝑑₇^∞_{𝑛=1𝐹3⊕Φ1=𝑛}𝑥²₈−1𝐹71³⊕Φ₁𝑞^𝑛+𝑑₈^∞_{𝑛=1Φ4=𝑛}𝑥²₁−9𝐹71²⊕Φ₂𝑞^𝑛+𝑑₉^∞_{𝑛=1Φ4=𝑛}𝑥²₂−1𝐹142²⊕Φ₂𝑞^𝑛+𝑑^∞_{10𝑛=1Φ4=𝑛}𝑥²₅−9𝐹142²⊕Φ₂𝑞^𝑛+𝑑^∞_{11𝑛=1Ψ4=𝑛}𝑥²₁−3Ψ71⁴𝑞^𝑛+𝑑^∞_{12𝑛=1Ψ4=𝑛}𝑥₁𝑥₂−1Ψ284⁴𝑞^𝑛+𝑑^∞_{13𝑛=1Ψ4=𝑛}𝑥²₂−3Ψ142⁴𝑞^𝑛+𝑑^∞_{14𝑛=1𝐹3⊕Ψ1=𝑛}𝑥²₁−9𝐹71³⊕Ψ₁𝑞^𝑛+𝑑^∞_{15𝑛=1𝐹2⊕Ψ2=𝑛}𝑥²₂−1𝐹142²⊕Ψ₂𝑞^𝑛+𝑑^∞_{16𝑛=1𝐹2⊕Ψ2=𝑛}𝑥²₆−3𝐹142²⊕Ψ₂𝑞^𝑛+𝑑^∞_{17𝑛=1Φ3⊕Ψ1=𝑛}𝑥²₁−9Φ71³⊕Ψ₁𝑞^𝑛+𝑑^∞_{18𝑛=1Φ3⊕Ψ1=𝑛}𝑥²₂−1Φ71³⊕Ψ₁𝑞^𝑛+𝑑^∞_{19𝑛=1Φ1⊕Ψ3=𝑛}𝑥²₂−1Φ71¹⊕Ψ₃𝑞^𝑛+𝑑^∞_{20𝑛=1Φ1⊕Ψ3=𝑛}𝑥²₄−3Φ142¹⊕Ψ₃𝑞^𝑛+𝑑^∞_{21𝑛=1𝐹3⊕Λ1=𝑛}𝑥²₁−9𝐹71³⊕Λ₁𝑞^𝑛+𝑑^∞_{22𝑛=1𝐹3⊕Λ1=𝑛}𝑥²₂−1𝐹142³⊕Λ₁𝑞^𝑛+𝑑^∞_{23𝑛=1Φ1⊕Λ3=𝑛}𝑥₁𝑥₂−1Φ284¹⊕Λ₃𝑞^𝑛+𝑑^∞_{24𝑛=1Φ1⊕Λ3=𝑛}𝑥²₂−1Φ71¹⊕Λ₃𝑞^𝑛+𝑑^∞_{25𝑛=1Φ1⊕Λ3=𝑛}𝑥²₄−2Φ71¹⊕Λ₃𝑞^𝑛+𝑑^∞_{26𝑛=1Ψ3⊕Λ1=𝑛}𝑥²₂−3Ψ142³⊕Λ₁𝑞^𝑛+𝑑^∞_{27𝑛=1𝐹2⊕Φ1⊕Ψ1=𝑛}𝑥²₅−9𝐹142²⊕Φ₁⊕Ψ₁𝑞^𝑛+𝑑^∞_{28𝑛=1𝐹2⊕Φ1⊕Ψ1=𝑛}𝑥₅𝑥₆−1𝐹284²⊕Φ₁⊕Ψ₁𝑞^𝑛+𝑑^∞_{29𝑛=1𝐹2⊕Φ1⊕Ψ1=𝑛}𝑥₇𝑥₈−1𝐹284²⊕Φ₁⊕Ψ₁𝑞^𝑛. (45)

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