Riemann hypothesis for odd period polynomials for newforms on Γ₀(N)

Abstract

Let k be an even integer and f ∈ S_k(Γ₀(N)) be a newform of weight k. We prove that for each integer N ≥ 16 and for k ≥ k_N,ε (depending on N), all of nonzero zeros of the odd period polynomial associated to the newform f are on the circle |z| = 1 / √N. For each integer 3 ≤ N ≤ 15, we also investigate the location of the zeros of the odd period polynomial associated to the newform f.