  A two-dimensional Schr\"odinger operator with a constant magnetic field
perturbed by a smooth compactly supported potential is considered. The spectrum
of this operator consists of eigenvalues which accumulate to the Landau levels.
We call the set of eigenvalues near the $n$'th Landau level an $n$'th
eigenvalue cluster, and study the distribution of eigenvalues in the $n$'th
cluster as $n\to\infty$. A complete asymptotic expansion for the eigenvalue
moments in the $n$'th cluster is obtained and some coefficients of this
expansion are computed. A trace formula involving the first eigenvalue moments
is obtained.
