Asymptotics of Hecke polynomial coefficients on the Atkin-Lehner eigenspaces

Let $S_k^{\sigma }(N)$ denote the space of cusp forms of level $N$, weight $k$, and Atkin-Lehner sign pattern $\sigma$, and $S_k^{\mathrm{n}\mathrm{e}\mathrm{w},\sigma }$ denote its new subspace. In this paper, we study the asymptotic behavior of the coefficients of the $m$-th Hecke polynomial over $S_k^{\sigma }(N)$ and $S_k^{\mathrm{n}\mathrm{e}\mathrm{w},\sigma }(N)$. In particular, we show that in certain settings, all but finitely many of these coefficients take a particular sign. We also study settings in which the coefficients do not tend to any particular sign.