The Hecke algebra \(T_k\) has large index (Q1882370)

Let \(\mathbb T_k(N)^{\mathrm{new}}\) denote the Hecke algebra acting on newforms of weight \(k\) and level \(N\). We prove that the power of \(p\) dividing the index of \(\mathbb T_k(N)^{\mathrm{new}}\) inside its normalisation grows at least linearly with \(k\) (for fixed \(N\)), answering a question of Serre. We also apply our method to give heuristic evidence towards recent conjectures of \textit{K. Buzzard}, Astérisque 298, 1--15 (2005; Zbl 1122.11025)] and Mazur.