Imaginary Hamiltonian

[Malamala]

Hello! The Hamiltonian for nuclear spin independent parity violation in atoms is given by: $$H_{PV} = Q_w\frac{G_F}{\sqrt{8}}\gamma_5\rho(r)$$ Here $Q_w$ is the weak charge of the nucleus (which is a scalar), $G_F$ is the Fermi constant and $\rho(r)$ is the nuclear density. From the papers I read, this operator is P-odd, T-even, and purely imaginary. I am having a hard time proving this. I managed to prove that it is P-odd. $\rho(r)$ is P-even, as it depends on the magnitude of $r$, so all I need to prove is that $\gamma_5$ is P-odd. For spinors, the parity operator is $\gamma_0$, and it can be easily shown that $\gamma_0\gamma_5\gamma_0^\dagger = -\gamma_5$. Hence the above Hamiltonian is P-odd. I am not sure how to prove it is T-even and imaginary. The first issue is that I am not sure what is the spinor space operator for the time inversion. Can someone help me with this? Thank you!