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

Abstract:

Bruinier and Ono classified cusp forms of half-integral weight \[ F(z):=\sum _{n=0}^{\infty }a(n)q^n\in S_{\lambda +\frac {1}{2}}(\Gamma _0(N),\chi )\cap \mathbb {Z}[[q]]\] whose Fourier coefficients are not well distributed for modulo odd primes $\ell$. 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.