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The determination of the electron neutrino mass by electron capture in 163Ho relies on a precise understanding of the deexcitation of a core hole after an electron-capture event. Here we present an ab initio calculation of the electron-capture spectrum of 163Ho, i.e., the 163Ho decay rate as a function of the energy distribution between the 163Dy daughter atom and the neutrino. Our current level of theory includes all intra-atomic decay channels and many-body interactions on a basis of fully relativistic bound orbitals. We use theoretical methods developed and extensively used for the calculation of core level spectroscopy on correlated electron materials. Our comparison to experimental electron-capture data critically tests the accuracy of these theories. We find that relativistic interactions beyond the Dirac equation lead to only minor shifts of the spectral peaks. The electronic relaxation after an electron-capture event due to the modified nuclear potential leads to a mixing of different edges, but, due to conservation of angular momentum of each scattered electron, no additional structures emerge. Many-body Coulomb interactions lead to the formation of multiplets and to additional peaks corresponding to multiple core holes created via Auger decay. Multiplets crucially change the appearance of the resonances on a Rydberg energy scale. The additional structures due to Auger decay are, although clearly visible, relatively weak compared to the single core hole states and are incidentally far away from the end-point region of the spectrum. As the end point of the spectrum is affected most by the neutrino mass, these additional states do not directly influence the statistics for determining the neutrino mass. The multiplet broadening and Auger shake-up of the main core-level edges do, however, change the apparent linewidth and accompanying lifetime of these edges. Fitting core-level edges, either in electron-capture spectroscopy or in x-ray absorption spectroscopy, by a single resonance thus leads to an underestimation of the core hole lifetime.