  Besides being one of the most fundamental basic issues of plasma physics, the
stability analysis of an electron beam-plasma system is of critical relevance
in many areas of physics. Surprisingly, decades of extensive investigation had
not yet resulted in a realistic unified picture of the multidimensional
unstable spectrum within a fully relativistic and kinetic framework. All
attempts made so far in this direction were indeed restricted to simplistic
distribution functions and/or did not aim at a complete mapping of the
beam-plasma parameter space. The present paper comprehensively tackles this
problem by implementing an exact linear model. We show that three kinds of
modes compete in the linear phase, which can be classified according to the
direction of their wavenumber with respect to the beam. We then determine their
respective domain of preponderance in a three-dimensional parameter space. All
these results are supported by multidimensional particle-in-cell simulations.
