Signs of the Second Coefficients of Hecke Polynomials

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