Clayton et al.'s nonvanishing conjecture for characteristic-polynomial coefficients

Let $m\geq1$ and let $k\geq2$ be even, and set $n=\dim S_k(\Gamma_0(1))$. Write the characteristic polynomial of $T_m(1,k)$ as

$$T_m(1,k)(x)=x^n-a_1(m,1,k)x^{n-1}+a_2(m,1,k)x^{n-2}-\ldots+(-1)^n a_n(m,1,k).$$

Clayton et al.'s nonvanishing conjecture. Then

$$a_i(m,1,k)\neq0\quad\text{for }1\leq i\leq n.$$

This conjecture asserts simultaneous nonvanishing of all coefficients of the characteristic polynomial of level-one Hecke operators, strengthening nonvanishing questions for the first and second coefficients; its resolution is not supplied here.