  We explore a notion of bent sequence attached to the data consisting of an
Hadamard matrix of order $n$ defined over the complex $q^{th}$ roots of unity,
an eigenvalue of that matrix, and a Galois automorphism from the cyclotomic
field of order $q.$ In particular we construct self-dual bent sequences for
various $q\le 60$ and lengths $n\le 21.$ Computational construction methods
comprise the resolution of polynomial systems by Groebner bases and eigenspace
computations. Infinite families can be constructed from regular Hadamard
matrices, Bush-type Hadamard matrices, and generalized Boolean bent
functions.As an application, we estimate the covering radius of the code
attached to that matrix over $\Z_q.$ We derive a lower bound on that quantity
for the Chinese Euclidean metric when bent sequences exist. We give the
Euclidean distance spectrum, and bound above the covering radius of an attached
spherical code, depending on its strength as a spherical design.
