We present a numerical investigation of the relativistic-field-theoretic (RFT) formalism, used to predict the discrete energy spectrum of three pions in a finite volume. Applying the generalization of [JHEP07(2020)047], we extract results for all non-maximal isospin values ($I_{\pi\pi\pi} = 2,1,$ and $0$), for different total momenta $\boldsymbol{P}$, and for various irreducible representations of the finite-volume symmetry group. We restrict attention to the unphysical scenario in which the three-particle interactions are set to zero. This set-up thus serves as a baseline for future lattice QCD calculations that will aim to extract such three-body interactions.