Looking for properties of, or formulae for eigenvalues of a symmetric matrix reminiscent of Toeplitz matrices

Question

I'm looking at $N\times N$ matrices $M_N$ of the form $$M_4=\begin{pmatrix}1 & a & a^2 & a^3 \\ a & 1 & a & a^2 \\ a^2 & a & 1 & a \\ a^3 & a^2 & a & 1\end{pmatrix},$$ where $a$ is a complex number of unit modulus. I'm particularly interested in large matrices with the property $a^N=1$. I've come up empty in several attempts trying to find eigenvalues or eigenvectors of this matrix analytically for sizes larger than 4, but the fact that this matrix only has one parameter and its close relation to Toeplitz matrices makes me hope someone has studied these things before.