MA342J: Introduction to Modular Forms
Homework 4, due on March 8
1. Prove that if f (z) ∈ Mk (Γ0 (M )) then f (N z) ∈ Mk (Γ0 (M N )).
Hint: Use questions 1 and 2 in tutorial 3.
2. For the Eisenstein series
E2 (z) = 1 − 24
X X d qn
n≥1
d|n
3
is equivariant under
the (nonholomorphic) function E2∗ (z) = E2 (z)− πIm(z)
SL(2, Z) in weight 2. (See the study week assignment.) Use this fact to
show that the functions
f1 = E2 (z) − 2E2 (2z) , f2 = E2 (2z) − 2E2 (4z) .
belong to M2 (Γ0 (4)). Write q-expansions for f1 and f2 and observe that
these modular forms are linearly independent.
Remark: Dimension formula shows that dim M2 (Γ0 (4)) = 2, hence you
constructed a basis for this space.
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