2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/231456The 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,χ}_{l,k} for all primes l, all weights k>1 and all characters χtaking 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.10 pagesNumber Theory11F33; 11F25, 11R29text