Exceptional Congruences for Eta-quotient newforms

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 $\hat{I}{“}_0(1)={\mathrm{SL}}_2(\mathbb{Z})$ 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 $S_k(N,\ddot{I}‡)$. When $k\ge 2$, we prove them using the theory of modular forms modulo primes. We also prove extensions of these congruences modulo prime powers.

18 pages, 2 figures. Comments are welcome!