On the parity of coefficients of eta powers

We consider a special subsequence of the Fourier coefficients of powers of the Dedekind $\eta$-function, analogous to the sequence $\delta_\ell := 24^{-1} \pmod{\ell}$ on which exceptional congruences of the partition function are supported. Therefrom we define a notion of density $D(r)$ for a normalized eta-power $\eta^r$ measuring the proportion of primes $\ell$ for which the order at infinity of $U_\ell (\eta^r)$ modulo 2 is maximal. We relate $D(r)$ to a notion of density measuring nonzero prime Fourier coefficients introduced by Bella\"iche, and use this to completely classify the vanishing of and establish upper bounds for $D(r)$. Furthermore, for several infinite families of $\eta$ powers corresponding to dihedral/CM mod-2 modular forms in the sense of Nicholas-Serre and Bella\"iche, we explicitly compute the densities $D$. We rely on Galois-theoretic techniques developed by Bella\"iche in level 1 and extend these to level 9. En passant we take the opportunity to communicate proofs of two of Bella\"iche's unpublished results on densities of mod-$2$ modular forms.