New results provided by numerical simulations of the weak instability of a warm electron beam in a collisionless plasma are presented. The theoretical model considers the self-consistent resonant interactions of beam particles with wave packets of broad spectra; it is derived using some of the initial approximations of the standard derivation of the quasilinear diffusion equation in the weak turbulence approach, without, however, the assumption of randomly phased waves. A huge number of particle trajectories calculated over long times by a symplectic code are analyzed using various statistical algorithms. The dynamics of the beam relaxation and the saturation of the wave spectrum are studied and compared with the analytical solutions provided by the quasilinear theory of weak turbulence. The most interesting results concern the presence of strong and persistent irregularities in the wave energy spectrum at saturation, which are linked to large velocity variations observed in the particles' dynamics and to non-Gaussian local diffusion. Quantitative estimates of the diffusion coefficients are given and compared with predictions of the weak turbulence theory.