$h_p(\omega^i,T)$ has $\mathbb{Z}_p$-coefficients

Let $p$ be a prime, $i$ an integer and $\gamma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Let $V_i$ be the $\omega^i$-eigenspace of $V = H_\infty \otimes_{\mathbb{Z}_p}\mathbb{Q}_p$, and let

Then all coefficients of $h_p(\omega^i, T)$ lie in $\mathbb{Z}_p$; that is, $h_p(\omega^i,T)$ is the image of a polynomial in $\mathbb{Z}_p[T]$.

This is needed for the source's statement to make sense: the main conjecture compares the ideal of $\mathbb{Z}_p[[T]]$ generated by $h_p(\omega^i, T)$ with the one generated by $G_p(\omega^{1-i}, T)$.

Formalization Note If $V_i$ were infinite-dimensional, charPoly would be $0$ by convention, and the statement holds trivially in that case.