¹⁶³Ho to ¹⁶³Dy is, probably due to the small Q value of about 2.5 keV, the best case to determine the neutrino mass by electron capture. The energy of the Q value is distributed between the excitation of dysprosium (and the neglected small recoil of holmium) and the relativistic energy of the emitted neutrino including the rest mass. The reduction of the upper end of the deexcitation spectrum of dysprosium below the Q values allows us to determine the neutrino mass. The excitation of dysprosium can be calculated in the sudden approximation of the overlap of the electron wave functions of holmium minus the captured electron and one-, two-, three-, and multiple hole excitations in dysprosium. Robertson [R. G. H. Robertson, Phys. Rev. C 91, 035504 (2015)] and Faessler and Simkovic [A. Faessler and F. Simkovic, Phys. Rev. C 91, 045505 (2015)]  have calculated the influence of the two-hole states on the dysprosium spectrum. Here for the first time the influence of the three-hole states on the deexcitation bolometer spectrum of ¹⁶³Dy  is presented. The electron wave functions and the overlaps are calculated self-consistently in a fully relativistic and antisymmetrized Dirac-Hartree-Fock approach in holmium and in dysprosium. The electron orbitals in Dy are determined including the one-hole states in the self-consistent iteration. The influence of the three-hole states on the Dy deexcitation (by x rays and Auger electrons) can hardly be seen. The three-hole states seem not to be relevant for the determination of the electron neutrino mass.