Ramanujan famously found congruences for the partition function like p(5n+4)
= 0 modulo 5. We provide a method to find all simple congruences of this type
in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is
non-vanishing on the upper half plane. This is applied to answer open questions
about the (non)-existence of congruences in the generating functions for
overpartitions, crank differences, and 2-colored F-partitions.