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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. 1