  For all odd primes N up to 500000, we compute the action of the Hecke
operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the
reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as
eigenvalues. We then partially explain the results in terms of class field
theory and modular mod-2 Galois representations. As a byproduct, we obtain some
nonexistence results on elliptic curves and modular forms with certain mod-2
reductions, extending prior results of Setzer, Hadano, and Kida.
