  We investigate spectral conditions on Hermitian matrices of roots of unity.
Our main results are conjecturally sharp upper bounds on the number of residue
classes of the characteristic polynomial of such matrices modulo ideals
generated by powers of $(1-\zeta)$, where $\zeta$ is a root of unity. We also
prove a generalisation of a classical result of Harary and Schwenk about a
relation for traces of powers of a graph-adjacency matrix, which is a crucial
ingredient for the proofs of our main results.
