On the irreducibility of Hecke polynomials

Let T_{n, k}(X) be the characteristic polynomial of the nth 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 ≥ 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 le; 4,000,000.