Chapter 10, Problem 10.70E

Assume that for a particle on a ring the operator for the angular momentum, $\widehat{p_{\phi }}$, is $-i\hslash \left(\delta /\delta \varphi \right)$. What is the eigenvalue for momentum for a particle having (unnormalized) $\Psi$ equal to $e^{3i\varphi }$? The integration limits are $0$ to $2\pi$. What is the average value of the momentum, $\langle p_{\phi }\rangle$ for a particle having this wavefunction? How are these results justified?