Zeros of Hecke Polynomials Arising From Weak Eigenforms

We attach Hecke polynomials [Formula: see text]([Formula: see text]) to weak Hecke eigenforms [Formula: see text] of weight 2 – [Formula: see text] and show that, for large [Formula: see text], every zero is simple and lies in [0, 1728]. The construction pulls back a weakly holomorphic Hecke combination of [Formula: see text] along [Formula: see text]; the analysis follows Hecke orbits on the unit-circle arc [Formula: see text], 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.