Distribution of the coefficients of modular forms and the partition function

Let ? be an odd prime and j, s be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo an odd positive integer M. As an application, we investigate the distribution of the ordinary partition function p(n) modulo ? ^j and prove that for each integer 1?≤ r?<?? ^j , $$\sharp\{1\le n\le X\ |\ p(n)\equiv r\pmod{\ell^j} \}\gg_{s,r,\ell^j} \frac{\sqrt X}{\log X}(\log\log X)^s.$$