  The multipole moments of stationary axially symmetric vacuum or electrovacuum
spacetimes can be expressed in terms of the power series expansion coefficients
of the Ernst potential on the axis. In this paper we present a simpler, more
efficient calculation of the multipole moments, applying methods introduced by
B\"ackdahl and Herberthson. For the non-vacuum electromagnetic case, our
results for the octupole and higher moments differ from the results already
published in the literature. The reason for this difference is that we correct
an earlier unnoticed mistake in the power series solution of the Ernst
equations. We also apply the presented method to directly calculate the
multipole moments of a 5-parameter charged magnetized generalization of the
Kerr and Tomimatsu-Sato exact solutions.
