Eigenvalues of a weird matrix

What are the eigenvalues of the following matrix, defined for $n\geq 3$:
$$A_n=\left(\begin{array}{llll} \cos (\log (n)) & \cos \left(\log \left(n^2\right)\right) & \cdots & \cos \left(\log \left(n^n\right)\right) \\ \sin (\log (n)) & \sin \left(\log \left(n^2\right)\right) & \cdots & \sin \left(\log \left(n^n\right)\right) \end{array}\right)$$