We performed microwave spectroscopy of an InAs nanowire between superconducting contacts implementing a finite-length, multichannel Josephson weak link. Certain features in the spectra, such as the splitting by spin-orbit interactions of the transition lines among Andreev states, have been already understood in terms of noninteracting models. However, we identify here additional transitions, which evidence the presence of Coulomb interactions. By combining experimental measurements and model calculations, we reach a qualitative understanding of these very rich Andreev spectra.