High-spin states of $^{40}$K have been populated in the fusion-evaporation reaction $^{12}$C($^{30}$Si,np)$^{40}$K and studied by means of $gamma$-ray spectroscopy techniques using one AGATA triple cluster detector, at INFN - Laboratori Nazionali di Legnaro. Several new states with excitation energy up to 8 MeV and spin up to $10^-$ have been discovered. These new states are discussed in terms of J=3 and T=0 neutron-proton hole pairs. Shell-model calculations in a large model space have shown a good agreement with the experimental data for most of the energy levels. The evolution of the structure of this nucleus is here studied as a function of excitation energy and angular momentum.
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