We determine the joint density of secular coefficients and of the traces of powers for random unitary matrices from Dyson's circular ensembles. A byproduct are workable `unitarity conditions', necessary and sufficient for all roots of a polynomial to be unimodular and different. These findings will be useful for combinations of random-matrix theory with semiclassical approximations in calculating spectral properties of quantum systems whose classical limit is globally chaotic. In a first such application we show that reasonable quasi-energy spectra result by throwing dice à la random-matrix theory to complement restricted knowledge (rigorous or from short periodic orbits) of the secular polynomial of a unitary Floquet matrix.