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MA342J: Introduction to Modular Forms Homework 5, due on March 28 1. Use the theorem stated in class to check that X 2 2 Θ(z) = q m +mn+n ∈ M1 (Γ0 (3), χ) m,n∈Z where χ(n) = −3 n 0 , 3 | n = 1 , n ≡ 1 mod 3 −1 , n ≡ 2 mod 3 is the nontrivial Dirichlet charater modulo 3. Also in this space we have the Eisenstein series ∞ E1,χ = 1 XX + χ(d) q n ∈ M1 (Γ0 (3), χ) . 6 n=1 d|n Use the fact that dim M1 (Γ0 (3), χ) = 1 to find the number of integer solutions (m, n) ∈ Z2 to the equation m2 + mn + n2 = 147 . 2. Describe all spherical homogeneous polynomials P (x, y) of degree 2 with respect to the quadratic form Q(m, n) = m2 + mn + n2 . Write X 2 2 ΘP (z) = P (m, n) q m +mn+n m,n∈Z in terms of the basis ∞ 1 XX χ(d)d2 q n , E3,χ = − + 9 n=1 d|n 0 E3,χ = ∞ X X n=1 d|n of the space M3 (Γ0 (3), χ). 1 χ(d) n 2 n q d