  We consider the semiclassical magnetic Laplacian $\mathcal{L}_h$ on a
Riemannian manifold, with a constant-rank and non-vanishing magnetic field $B$.
Under the localization assumption that $B$ admits a unique and non-degenerate
well, we construct three successive Birkhoff normal forms to describe the
spectrum of $\mathcal{L}_h$ in the semiclassical limit $\hbar \rightarrow 0$.
We deduce an expansion of all the eigenvalues under a threshold, in powers of
$\hbar^{1/2}$.
