On the Zeros of Certain Cusp Forms

Abstract

F.K.C. Rankin and H.P.F. Swinnerton–Dyer proved that all the zeros of the Eisenstein Series $E_k$ contained in the standard fundamental domain $\mathcal{F}$ lie on the arc $A\,{=}\, \{ e^{i\theta}| {\pi}/{3} \,{\le}\, \theta \,{\le}\, {\pi}/{2}\}$. Recently, J. Getz has generalized the method of Rankin and Swinnerton–Dyer to show that modular forms under certain conditions have similar properties. In this paper we prove similar results for certain types of cusp forms, motivated by the work of R.A. Rankin. Further, we give a closed formula for the zeros of a class of cusp forms in terms of the Fourier coefficients following the method of Kohnen.