Let $C = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$ be a $3 \times 3$ permutation matrix representing a cyclic shift. If $\omega = e^{\frac{2\pi i}{3}}$, what are the eigenvalues of $C$?