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

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.