We present three-dimensional magnetohydrodynamic (MHD) numerical simulations of the evolution of self-gravitating and weakly magnetized disks with an adiabatic equation of state. Such disks are subject to the development of both the magnetorotational and gravitational instabilities, which transport angular momentum outward. As in previous studies, our hydrodynamic simulations show the growth of a strong m=2 spiral structure. This spiral disturbance drives matter toward the central object and disappears when the Toomre parameter, Q, has increased well above unity. When a weak magnetic field is present as well, the magnetorotational instability grows and leads to turbulence. In that case, the strength of the gravitational stress tensor is lowered by a factor of ~2 compared with the hydrodynamic run and oscillates periodically, reaching very small values at its minimum. We attribute this behavior to the presence of a second spiral mode with higher pattern speed than the one that dominates in the hydrodynamic simulations. It is apparently excited by the high-frequency motions associated with MHD turbulence. The nonlinear coupling between these two spiral modes gives rise to a stress tensor that oscillates with a frequency that is a combination of the frequencies of each of the modes. This interaction between MHD turbulence and gravitational instabilities therefore results in a smaller mass accretion rate onto the central object.