Parity of the coefficients of certain eta-quotients, III: The case of pure eta-powers

We continue a program of investigating the parity of the coefficients of eta-quotients, with the goal of elucidating the parity of the partition function. In this paper, we consider positive integer powers t of the Dedekind eta-function. Previous work and conjectures suggest that arithmetic progressions in which the Fourier coefficients of these functions are even should be numerous. We explicitly identify infinite classes of such progressions modulo prime squares for several values of t, and we offer two broad conjectures concerning their existence in general.