Nonvanishing of second coefficients of Hecke polynomials on the newspace

For [Formula: see text], let [Formula: see text] be coprime to m, [Formula: see text], and [Formula: see text] be a Dirichlet character modulo N with [Formula: see text]. Then let [Formula: see text] denote the restriction of the mth Hecke operator to the space [Formula: see text]. We demonstrate that for fixed m and trivial character [Formula: see text], the second coefficient of the characteristic polynomial of [Formula: see text] vanishes for only finitely many pairs [Formula: see text], and we further determine the sign. To demonstrate our method, for [Formula: see text], we also compute all pairs [Formula: see text] for which the second coefficient vanishes. In the general character case, we also show that excluding an infinite family where [Formula: see text] is trivial, the second coefficient of the characteristic polynomial of [Formula: see text] vanishes for only finitely many triples [Formula: see text].