The $Xi_c^+ K^-$ mass spectrum is studied with a sample of $pp$ collision data corresponding to an integrated luminosity of 3.3 fb$^{-1}$, collected by the LHCb experiment. The $Xi_c^+$ is reconstructed in the decay mode $p K^- pi^+$. Five new, narrow excited $Omega_c^0$ states are observed: the $Omega_c(3000)^0$, $Omega_c(3050)^0$, $Omega_c(3066)^0$, $Omega_c(3090)^0$, and $Omega_c(3119)^0$. Measurements of their masses and widths are reported.
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