On the Zeros and Coefficients of Certain Weakly Holomorphic Modular Forms

For this paper we assume familiarity with the basics of the theory of modular forms as may be found, for instance, in Serre’s classic introduction [12]. A weakly holomorphic modular form of weight k ∈ 2Z for Γ = PSL2(Z) is a holomorphic function f on the upper half-plane that satisfies f( cτ+d ) = (cτ + d)f(τ) for all ( a b c d ) ∈ Γ and that has a q-expansion of the form f(τ) = ∑ n≥n0 a(n)q , where q = e and n0 = ord∞(f). Such an f is holomorphic if n0 ≥ 0 and a cusp form if n0 ≥ 1. Let Mk denote the vector space of all weakly holomorphic modular forms of weight k. Any nonzero f ∈ Mk satisfies the valence formula