  In this article, we investigate inertial modes of rigidly rotating neutron
stars, i.e. modes for which the Coriolis force is dominant. This is done using
the assumption of a fixed spacetime (Cowling approximation). We present
frequencies and eigenfunctions for a sequence of stars with a polytropic
equation of state, covering a broad range of rotation rates. The modes were
obtained with a nonlinear general relativistic hydrodynamic evolution code. We
further show that the eigenequations for the oscillation modes can be written
in a particularly simple form for the case of arbitrary fast, but rigid
rotation. Using these equations, we investigate some general characteristics of
inertial modes, which are then compared to the numerically obtained
eigenfunctions. In particular, we derive a rough analytical estimate for the
frequency as a function of the number of nodes of the eigenfunction, and find
that a similar empirical relation matches the numerical results with unexpected
accuracy. We investigate the slow rotation limit of the eigenequations,
obtaining two different sets of equations describing pressure and inertial
modes. For the numerical computations we only considered axisymmetric modes,
while the analytic part also covers nonaxisymmetric modes. The eigenfunctions
suggest that the classification of inertial modes by the quantum numbers of the
leading term of a spherical harmonic decomposition is artificial in the sense
that the largest term is not strongly dominant, even in the slow rotation
limit. The reason for the different structure of pressure and inertial modes is
that the Coriolis force remains important in the slow rotation limit only for
inertial modes. Accordingly, the scalar eigenequation we obtain in that limit
is spherically symmetric for pressure modes, but not for inertial modes.
