Coefficients of half-integral weight modular forms modulo ℓ^j

Suppose that ℓ ≥ 5 is prime, that j > 0 is an integer, and that F(z) is a half-integral weight modular form with integral Fourier coefficients. We give some general conditions under which the coefficients of F are "well-distributed" modulo ℓ^j. As a consequence, we settle many cases of a classical conjecture of Newman by proving, for each prime power ℓ^j with ℓ ≥ 5, that the ordinary partition function p(n) takes each value modulo ℓ^j infinitely often.