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The electron beam generated in a self-modulated laser-wakefield accelerator is characterized in detail. A transverse normalized emittance of 0.06 π mm mrad, the lowest ever for an electron injector, was measured for 2 MeV electrons. The electron beam was observed to have a multicomponent beam profile and energy distribution. The latter also undergoes discrete transitions as the laser power or plasma density is varied. In addition, dark spots that form regular modes were observed in the electron beam profile. These features are explained by analysis and test particle simulations of electron dynamics during acceleration in a three-dimensional plasma wakefield.