Eigenvalues of circulant like matrix

Question

Let $q$ be a positive integer and let $P$ be the $p \times p$ permutation matrix given by $P(e_i)=e_{i+q}$, where the indices are mod $p$. Let $v$ be a column vector. What are, and how to compute, the eigenvalues of the matrix $M=[v,Pv,P^2v,\dots , P^{p-1}v]$? An answer with $p$ prime and $q \leq p$ would be great.