Riemann hypothesis for odd period polynomials for newforms on $\Gamma_0(N)$

Let $k$ be an even integer and $f \in S_k(\Gamma_0(N))$ be a newform of weight $k$. We prove that for each integer $N\geq16$ and for $k\geq k_{N,\epsilon}$ (depending on $N$), all of nonzero zeros of the odd period polynomial associated to the newform $f$ are on the circle $|z|=1/\sqrt{N}$. For each integer $3\leq N\leq 15$, we also investigate the location of the zeros of the odd period polynomial associated to the newform $f$.