An algebraic interpretation of the gauss sums

Let be a finite arithmetic sequence in C with elements the values of a Dirichlet character Ξ mod n. If Xis the circulant n × n matrix with elements Ξ(i) then the eigenvalues of X are the Gauss sums that correspond to Ξ. Moreover, if Ξ=Ξ1 is the principal Dirichlet character mod n, then the eigenvalues of X are the Ramanujan sums C n(K).