  The behaviour of Hecke polynomials modulo p has been the subject of some
study. In this note we show that, if p is a prime, the set of integers N such
that the Hecke polynomials T^{N,\chi}_{l,k} for all primes l, all weights k>1
and all characters \chi taking values in {+1,-1} splits completely modulo p has
density 0, unconditionally for p=2 and under the Cohen-Lenstra heuristics for
odd p. The method of proof is based on the construction of suitable dihedral
modular forms.
