Math 7711-01 Modular forms: Homework set 4 1. Using Poisson summation formula, show that if τ ∈ H and ∞ X 2 θ(τ ) = e2πin τ , n=−∞

then the following transformation is valid:   √ 1 = −2iτ θ(τ ). θ − 4τ 2. Show that the map Γ1 (N ) → Z/N Z given by   a b 7→ b (mod N ) c d surjects and has kernel Γ(N ). 3. Show that the map Γ0 (N ) → (Z/N Z)∗ given by   a b 7→ d (mod N ) c d surjects and has kernel Γ1 (N ). 4. Show that

 Y 1 [SL2 (Z) : Γ0 (N )] = N 1+ . p p|N

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