  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
show that if there exists n>1 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
all primes p.
