  For $k\ge 2$ even, let $d_{k,N}$ denote the dimension of the largest simple
Hecke submodule of $S_{k}(\Gamma_0(N); \mathbb{Q})^\text{new}$. We show, using
a 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 were
only 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.
