A note on the dimension of the largest simple Hecke submodule

For $k\ge 2$ even, let $d_{k,N}$ denote the dimension of the largest simple Hecke submodule of $S_k({\mathrm{\Gamma }}_0(N);\mathbb{Q}{)}^{\text{new}}$. We show, using a simple analytic method, that $d_{k,N}{\gg }_k\log \log N\mathrm{/}\log (2p)$ 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 $S_k({\mathrm{\Gamma }}_0(N),\chi )$, with $k \geq 2$anintegerand$\chi$ an arbitrary nebentypus.