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For $k\ge 2$ even, let $d_{k,N}$ denote the dimension of the largest simpleHecke submodule of $S_{k}(\Gamma_0(N); \mathbb{Q})^\text{new}$. We show, usinga simple analytic method, that $d_{k,N} \gg_k \log\log N / \log(2p)$ with $p$the smallest prime co-prime to $N$. Previously, bounds of this quality wereonly known for $N$ in certain subsets of the primes. We also establish similar(and sometimes stronger) results concerning $S_{k}(\Gamma_0(N), \chi)$, with $k\geq 2$ an integer and $\chi$ an arbitrary nebentypus.

A note on the dimension of the largest simple Hecke submodule