On the Fourier coefficients of Modular forms of Half integral weight

Abstract

It is known that if the Fourier coefficients a(n) (n >= 1) of an elliptic modular form of even integral weight k >= 2 on the Hecke congruence subgroup Gamma(0)(N) (N is an element of N) satisfy the bound a(n) << (f) n(c) for all n >= 1, where c > 0 is any number strictly less than k - 1, then f must be cuspidal. Here we investigate the case of half-integral weight modular forms. The main objective of this note is to show that to deduce that f is a cusp form, it is sufficient to impose a suitable growth condition only on the Fourier coefficients a(vertical bar D vertical bar) where D is a fundamental discriminant with (- 1)(k) D > 0.