The integer group determinants for $\mathbb{Z}_p^n$

We give new conditions on allowable powers of a prime $p$ dividing an integer group determinant for the group $\mathbb{Z}_p^n$, and show these conditions are sharp for $n\leq 5$. The values coprime to $p$ for these groups are already known; determining which multiples of allowable powers of $p$ occur is much more complicated. We show that all multiples of sufficiently large powers of $p$ occur as integer group determinants of $\mathbb{Z}_p^n$. We also provide a complete characterization for the group $\mathbb{Z}_3^3$, where particular arithmetic conditions are required for multiples of certain powers of $3$.