On mod p representations which are defined over F_p: II

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.