  For $m \geq 1$, let $N \geq 1$ be coprime to $m$, $k \geq 2$, and $\chi$ be a
Dirichlet character modulo $N$ with $\chi(-1)=(-1)^k$. Then let
$T_m^{\text{new}}(N,k,\chi)$ denote the restriction of the $m$-th Hecke
operator to the space $S_k^{\text{new}}(\Gamma_0(N), \chi)$. We demonstrate
that for fixed $m$ and trivial character $\chi$, the second coefficient of the
characteristic polynomial of $T_m^{\text{new}}(N,k)$ vanishes for only finitely
many pairs $(N,k)$, and we further determine the sign. To demonstrate our
method, for $m=2,4$, we also compute all pairs $(N,k)$ for which the second
coefficient vanishes. In the general character case, we also show that
excluding an infinite family where $S_k^{\text{new}}(\Gamma_0(N), \chi)$ is
trivial, the second coefficient of the characteristic polynomial of
$T_m^{\text{new}}(N,k,\chi)$ vanishes for only finitely many triples
$(N,k,\chi)$.
