Exceptional Congruences for Eta-quotient newforms

TL;DR

This paper classifies specific congruences of eta-quotient newforms, extending previous work on modular form coefficients, and proves these congruences modulo primes and prime powers using modular forms theory.

Contribution

It provides a comprehensive classification of Type I and Type II congruences for eta-quotient newforms, extending Swinnerton-Dyer's classical results to new contexts.

Findings

Classified congruences for eta-quotient newforms in various weights.

Extended congruences to prime power moduli.

Applied modular forms modulo primes to prove these results.

Abstract

In 1973, Swinnerton-Dyer completely classified all congruences for coefficients of normalized eigenforms in weights $k \in \{12, 16, 18, 20, 22, 26\}$ on $\Gamma_{0}(1) = \operatorname{SL}_{2}(\mathbb{Z})$ using the theory of modular Galois representations. In this paper, we classify congruences of Type I and Type II considered by Swinnerton-Dyer for the coefficients of eta-quotient newforms in $S_{k}(N, \chi)$. When $k \geq 2$, we prove them using the theory of modular forms modulo primes. We also prove extensions of these congruences modulo prime powers.