  The study of arithmetic properties of coefficients of modular forms $f(\tau)
= \sum a(n)q^n$ has a rich history, including deep results regarding
congruences in arithmetic progressions. Recently, work of C.-S. Radu, S.
Ahlgren, B. Kim, N. Andersen, and S. L\"{o}brich have employed the
$q$-expansion principle of P. Deligne and M. Rapoport in order to determine
more about where these congruences can occur. Here, we extend the method to
give additional results for a large class of modular forms. We also give
analogous results for generalized Frobenius partitions and the two mock theta
functions $f(q)$ and $\omega(q).$
