The influence of carrier density on magnetism in a zigzag graphene nanoribbon is studied in a $\pi$-orbital Hubbard-model mean-field approximation. Departures from half-filling alter the magnetism, leading to states with charge density variation across the ribbon and parallel spin-alignment on opposite edges. Finite carrier densities cause the spin-density near the edges to decrease steadily, leading eventually to the absence of magnetism. At low doping densities the system shows a tendency to multiferroic order in which edge charges and spins are simultaneously polarized.