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 ∈ 𝑆_𝑘(Γ₀(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.