Detecting large simple rational Hecke modules for $\Gamma_0(N)$ via congruences

We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_2(\Gamma_0(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_0(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. For prime levels $N\equiv 7\mod 8$ our method yields an unconditional bound of $d\ge\log_2\log_2(N/8)$, improving the known bound of $d\gg\sqrt{\log\log N}$ due to Murty--Sinha and Royer. We also discuss conditional bounds, the strongest of which is $d\gg_\epsilon N^{1/2-\epsilon}$ over a large set of primes $N$, contingent on Soundararajan's heuristics for the class number problem and Artin's conjecture on primitive roots. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_k(\mathrm{SL}_2(\mathbb{Z});\mathbb{Q})$.