The flavor evolution of neutrinos emitted by a supernova (SN) core is strongly affected by the refractive effects associated with the neutrino-neutrino interactions in the deepest stellar regions. Till now, all numerical studies have assumed the axial symmetry for the ``multi-angle effects{''} associated with the neutrino-neutrino interactions. Recently, it has been pointed out in Raffelt, Sarikas, and Seixas {[}Phys. Rev. Lett. 111, 091101 (2013)] that if this assumption is removed, a new multi-azimuthal-angle (MAA) instability emerges in the flavor evolution of the dense SN neutrino gas, in addition to the one caused by multi-zenith-angle effects. Inspired by this result, for the first time we numerically solve the nonlinear neutrino propagation equations in SN, introducing the azimuthal angle as an angular variable in addition to the usual zenith angle. We consider simple energy spectra with an excess of nu(e) over (nu) over bar (e). We find that even starting with a complete axial symmetric neutrino emission, the MAA effects would lead to significant flavor conversions in normal mass hierarchy, in cases otherwise stable under the only multi-zenith-angle effects. The final outcome of the flavor conversions, triggered by the MAA instability, depends on the initial asymmetry between nu(e) and (nu) over bar (e) spectra. If it is sufficiently large, final spectra would show an ordered behavior with spectral swaps and splits. Conversely, for small flavor asymmetries flavor decoherence among angular modes develops, also affecting the flavor evolution in the inverted mass hierarchy.