  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 $(N,k)$.
