Interlacing of zeros of period polynomials

Abstract

By a lot of previous work, it is known that the zeros of the period polynomial for a newform $f\in S_k({\mathrm{\Gamma }}_0(N))$ all lie on the circle $|z|=1/\sqrt{N}$. In this paper we show that these zeros satisfy various interlacing properties for fixed $N$ and varying $k$ when either $k$ or $N$ is large. We also present a complete result when $N=1$. Lastly, we establish the interlacing properties when $k$ is fixed and $N$ varies.