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arXiv · 2511.16039

Exceptional Congruences for Eta-quotient newforms

Eddie O'Sullivan·Henry Stone·Swati·Xiaolan Jin
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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.

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Eddie O'Sullivan, Henry Stone, Swati, Xiaolan Jin. 2025-11-20. Exceptional Congruences for Eta-quotient newforms. https://arxiv.org/abs/2511.16039

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