Coefficients of half-integral weight modular forms modulo lj

Suppose that l greater than or equal to 5 is prime, that j greater than or equal to 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 greater than or equal to 5, that the ordinary partition function p(n) takes each value modulo l(j) infinitely often.