On the Vanishing of Fourier Coefficients of Certain Genus Zero Newforms

Abstract

Abstract. Given a classical modular form f (z), a basic question is whether any of its Fourier coefficients vanish. This question remains open for certain modular forms. For example, let ∆(z) = τ (n)q n ∈ S 12 (Γ 0 (1)). A well-known conjecture of Lehmer asserts that τ (n) = 0 for all n. In recent work, Ono constructed a family of polynomials A n (x) ∈ Q[x] with the property that τ (n) vanishes if and only if A n (0) and A n (1728) do. In this paper, we establish a similar criterion for the vanishing of coefficients of certain newforms on genus zero groups of prime level.