On the parity of coefficients of eta powers

TL;DR

This paper studies the distribution of primes related to eta function coefficients, classifies when these coefficients vanish, and computes densities for specific eta powers using Galois theory and modular forms.

Contribution

It introduces a new density measure for eta power coefficients, classifies their vanishing behavior, and explicitly computes densities for families linked to dihedral and CM modular forms.

Findings

Classified vanishing of eta power coefficients based on prime densities.

Established upper bounds for the density D(r).

Computed densities D for specific eta powers related to modular forms.

Abstract

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 En passant we take the opportunity to communicate proofs of two of Bella\"iche's unpublished results on densities of mod-$2$ modular forms.