Complex equiangular Parseval frames and Seidel matrices containing $p$th roots of unity

We derive necessary conditions for the existence of complex Seidel matrices containing $p$th roots of unity and having exactly two eigenvalues, under the assumption that $p$ is prime. The existence of such matrices is equivalent to the existence of equiangular Parseval frames with Gram matrices whose off-diagonal entries are a common multiple of the $p$th roots of unity. Explicitly examining the necessary conditions for $p=5$ and $p=7$ rules out the existence of many such frames with a number of vectors less than 50, similar to previous results in the cube roots case. On the other hand, we confirm the existence of $p^2\times p^2$ Seidel matrices containing $p$th roots of unity, and thus the existence of the associated complex equiangular Parseval frames, for any $p\ge 2$. The construction of these Seidel matrices also yields a family of previously unknown Butson-type complex Hadamard matrices.