  In this paper we construct a Birkhoff normal form for a semiclassical
magnetic Schr{\"o}dinger operator with non-degenerate magnetic field, and
discrete magnetic well, defined on an even dimensional riemannian manifold M.
We use this normal form to get an expansion of the first eigenvalues in powers
of h^{1/2}, and semiclassical Weyl asymptotics for this operator.
