This study introduces a basis of states which are dressed by the interaction between three, largely populated, modes of Bogoliubov quasi-particles in a Bose-Einstein condensate at zero temperature. The dressed basis is obtained by diagonalizing the N×M matrix, which represents the coupling between mode k with N excitations and mode q with M excitations. The degeneracy between the excitation Fock states is removed by the interaction. The new dressed basis spans a spectrum of energies which is symmetric around E=0. The spectrum appears to be roughly linear, however, due to the non-linearity of the problem, energy differences between each pair of dressed states are slightly different.