A complex Hadamard matrix of order $N$ is a square matrix $A = [a_{ij}]$ of size $N$ with entries $|a_{ij}| = 1$ for all $i,j$; and further that $A^\dagger A= N I$ (or $A/\sqrt{N} $ is a unitary matrix).
Further, a complex Hadamard matrix of Butson-type is such that $a_{ij}$ is the $q$-th root of unity for some positive integer $q$.
My question is: when is the spectrum of the unitary associated with a Buston-type Hadamard matrix also comprised of $k$-th roots of unity? What conditions guarantee this / what can be checked?
For example, the Fourier matrix $F_{jk} = e^{i 2\pi (j-1)(k-1)/N}$ of order $N$ is Butson-type Hadamard and the unitary $F/\sqrt{N}$ has eigenvalues $\pm 1, \pm i$ (so fourth root of unity).