Modular forms of half-integral weight with few non-vanishing coefficients modulo $\ell $

Abstract

Bruinier and Ono classified cusp forms of half-integral weight F(z):=∞Σn=0a(n)q n ∈ S λ+1 2 (Γ 0 (N), Χ )∩Z[[q]] whose Fourier coefficients are not well distributed for modulo odd primes l. Ahlgren and Boylan established bounds for the weight of such a cusp form and used these bounds to prove Newman's conjecture for the partition function for prime-power moduli. In this note, we give a simple proof of Ahlgren and Boylan's result on bounds of cusp forms of half-integral weight.