Proper circulant weighing matrices CW(q^3-1, q^2) for every odd prime power q

For every odd prime power q we give an explicit proper circulant weighing matrix of order q^3-1 and weight q^2. Its first row is the sum of η(-N(u)) [(1-u)/(2u)] over the elements u of trace 1 in F_{q^3}, where N is the norm to F_q and η the quadratic character of F_q. Together with the known construction for even q, this shows that a proper CW(q^3-1, q^2) exists for every prime power q, as Arasu, Gordon and Zhang suggested. The case q=7 gives a proper CW(342,49), a parameter pair recorded as nonexistent in the La Jolla Circulant Weighing Matrix Repository, and q=9 gives a proper CW(728,81), recorded there as open. The proof uses the geometry of conics in PG(2,q) and Gauss and Jacobi sums.The archive contains the first rows for all odd q ≤ 27 and the programs that build and check them.