  The period polynomial $r_f(z)$ for a weight $k \geq 3$ newform $f \in
S_k(\Gamma_0(N),\chi)$ is the generating function for special values of
$L(s,f)$. The functional equation for $L(s, f)$ induces a functional equation
on $r_f(z)$. Jin, Ma, Ono, and Soundararajan proved that for all newforms $f$
of even weight $k \ge 4$ and trivial nebetypus, the "Riemann Hypothesis" holds
for $r_f(z)$: that is, all roots of $r_f(z)$ lie on the circle of symmetry $|z|
=1/\sqrt{N}$. We generalize their methods to prove that this phenomenon holds
for all but possibly finitely many newforms $f$ of weight $k \ge 3$ with any
nebentypus. We also show that the roots of $r_f(z)$ are equidistributed if $N$
or $k$ is sufficiently large.
