An analysis of electrostatic eigenmodes of a cylindrical metallic waveguide completely filled with a magnetized relativistic electron beam is presented. Self-fields are taken into account by considering a rotating electron beam. For the equilibrium configuration when this rotational motion is not too large, one can see a non-relativistic electron beam in the beam frame moving with the axial beam velocity. Therefore, electrostatic approximation can be assumed which fully incorporates beam space-charge effects, but neglects any fast electromagnetic processes in the beam frame. The resulting dispersion relation (DR) for the electrostatic waves that is Lorentz transformed back to the laboratory frame reveals the relativistic effects as electromagnetic fields of the wave. The resulting DR reduces to the previously found results in the absence of self-fields as well as in the non-relativistic limit. A numerical analysis is performed to study the dispersion characteristics of the space-charge and cyclotron modes. It was found that increasing the axial velocity lowers the asymptotic frequencies at infinite wave numbers for both electrostatic modes and the cutoff frequencies of the cyclotron modes. The effect of self-fields on the DR of the cyclotron modes is large and shifts the dispersion curves to lower frequencies without changing their shape appreciably.