Zeros of Hecke polynomials arising from weak eigenforms

We attach Hecke polynomials $P_n(F;x)$ to weak Hecke eigenforms $F$ of weight $2-k$ and show that, for large $n$, every zero is simple and lies in $[0,1728]$. The construction pulls back a weakly holomorphic Hecke combination of $F$ along $j$; the analysis follows Hecke orbits on the unit-circle arc $\mathcal{A}$, isolating a dominant “cosine” term and controlling the tail via Maass-Poincaré series and Whittaker/Bessel bounds. This extends the Rankin–Swinnerton-Dyer/Asai–Kaneko–Ninomiya picture from holomorphic forms to a broad class of harmonic Maass forms and yields a clean degree-monicity formula and simple criteria for zeros at $0$ and $1728$.