On the irreducibility of Hecke polynomials

Let $T_{n, k}(X)$ be the characteristic polynomial of the $n$th Hecke operator acting on the space of cusp forms of weight $k$ for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials $T_{n, k}(X)$. Using this criterion with some machine computation, we show that if there exists $n\geq 2$ such that $T_{n, k}(X)$ is irreducible and has the full symmetric group as Galois group, then the same is true of $T_{p, k}(X)$ for each prime $p\leq 4,000,000$.