Signs of the Second Coefficients of Hecke Polynomials

Let Tₘ(N, k, χ) be the m-th Hecke operator of level N, weight k ≥ 2, and nebentypus χ, where N is coprime to m. We first show that for any given m ≥ 1, the second coefficient of the characteristic polynomial of Tₘ(N, k, χ) is nonvanishing for all but finitely many triples (N,k,χ). Furthermore, for χ trivial and any fixed m, we determine the sign of the second coefficient for all but finitely many pairs $(N,k)$. Finally, for χ trivial and $m=3,4$, we compute the sign of the second coefficient for all pairs $(N,k)$.