On mod p modular representations which are defined over \F_p
  In this paper, we use techniques of Conrey, Farmer and Wallace to find spaces of modular forms $S_k(\Gamma_0(N))$ where all of the eigenspaces have Hecke eigenvalues defined over $\F_p$, and give a heuristic indicating that these are all such spaces. 
L. J. P. Kilford