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We study the circulant complex Hadamard matrices of order $n$ whose entriesare $l$-th roots of unity. For $n=l$ prime we prove that the only such matrix,up to equivalence, is the Fourier matrix, while for $n=p+q,l=pq$ with $p,q$distinct primes there is no such matrix. We then provide a list of equivalenceclasses of such matrices, for small values of $n,l$.

Small circulant complex Hadamard matrices of Butson type