Math 597 - Homework 2
1. (a) Let f(z) = η(z)8η(2z)8. Show that the map Φ : Mk(Γ0(2)) →Sk+8(Γ0(2)) given by
φ(g) = fg is an isomorphism.
(b) Use the fact that dim M2(Γ0(2)) = 1 to find a formula for dimMk(Γ0(2)) for all k.
2. Let
f(z) = η(8z)8
η(4z)2η(16z)2=
∞
X
n=1
a(n)qn.
(a) Show that f(z)∈S2(Γ0(64)). [You may wish to look up Ken’s book. It gives a formula for
the order of vanishing of an eta-quotient at each cusp].
(b) Show that if gcd(m, n) = 1 then a(mn) = a(m)a(n), and
a(pn) = a(p)a(pn−1)−pa(pn−2) if n≥2, p 6= 2,
and a(2n) = 0.
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