Congruences for modular forms and applications to crank functions

Motivated by the work of Mahlburg, which refined the work of Ono, we find congruences for a large class of modular forms. Moreover, we generalize the generating function of the Andrews–Garvan–Dyson crank of partitions and establish several new infinite families of congruences. In this framework, we show that both the birank of an ordered pair of partitions introduced by Hammond and Lewis, and $k$-crank of $k$-colored partitions introduced by Fu and Tang, have the same properties as the partition function and crank.