On the irreducibility of Hecke polynomials

Abstract

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 < 4,000,000.