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We investigate spectral conditions on Hermitian matrices of roots of unity.Our main results are conjecturally sharp upper bounds on the number of residueclasses of the characteristic polynomial of such matrices modulo idealsgenerated by powers of $(1-\zeta)$, where $\zeta$ is a root of unity. We alsoprove a generalisation of a classical result of Harary and Schwenk about arelation for traces of powers of a graph-adjacency matrix, which is a crucialingredient for the proofs of our main results.

Hermitian matrices of roots of unity and their characteristic polynomials