  Plane wave solutions to the cubic nonlinear Schr\"odinger equation on a torus
have recently been shown to behave orbitally stable. Under generic
perturbations of the initial data that are small in a high-order Sobolev norm,
plane waves are stable over long times that extend to arbitrary negative powers
of the smallness parameter. The present paper studies the question as to
whether numerical discretizations by the split-step Fourier method inherit such
a generic long-time stability property. This can indeed be shown under a
condition of linear stability and a non-resonance condition. They can both be
verified if the time step-size is restricted by a CFL condition in the case of
a constant plane wave. The proof first uses a Hamiltonian reduction and
transformation and then modulated Fourier expansions in time. It provides
detailed insight into the structure of the numerical solution.
