  A computational study of the electron paramagnetic resonance (EPR)
$g$-tensors and hyperfine tensors in Mg and Zn doped $\beta$-Ga$_2$O$_3$ is
presented. While Mg has been found previously to prefer the octahedral site, we
find here that Zn prefers the tetrahedral substitutional site. The EPR
signatures are found to be distinct for the two sites. Good agreement with
experiment is found for the $g$-tensor and hyperfine interaction for
Mg$_\mathrm{Ga2}$ and predictions are made for the Zn case.
