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