In this work the electronic properties of the MnCoSi half-Heusler alloy are evaluated within the formalism of the electron density functional. The calculations are made using ABINIT code in two approaches. They are based on the projector augmented waves (PAW) and on the norm-conserving pseudopotentials (NCPP). The hybrid exchange-correlation functional PBE0, which allows to remove partly the self-interaction of strongly correlated 3$d$ electrons of Co and Mn atoms, is implemented on the PAW basis. The electronic energy spectra of the MnCoSi crystal in the magnetic and nonmagnetic phases have been evaluated in the PAW basis states. In the process of optimizing the crystal structure, the interpolants of total energy $E(V)$ and pressure $P(V)$ as the functions of volume, and $V(P)$ as a function of pressure are found. Based on these dependencies, the bulk modulus $B_0$ = 196.1 GPa is found. As established, in the magnetic state the MnCoSi material is a metal for spin up and a semiconductor for spin down. In the non-magnetic state, the material is a metal. Electronic properties are obtained for the uncompressed and compressed states of the crystal. The pressure dependencies of the magnetic moment of atoms are also found. Within the second approach (NCPP), the elastic constants $c_{11}$, $c_{12}$, and $c_{44}$ are calculated, from which the moduli of elasticity, a sound waves velocities, and Debye temperature are derived. The bulk modulus $B_0$ = 197.1 GPa found in this approach is well compared with one obtained on the PAW basis.