Let $p\in\mathbb{N}$ be a prime and let us denote $N=p^k$ for some $k\in\mathbb{N}$. Let $\alpha$ be the larger root of the qudratic $q(x)=x^2+ax+1$ where $a\in 2\mathbb{Z}$ and $a^2>4$. Consider the ring $R=\mathbb{Z}[\alpha]$. How many elements are there in the quotient ring $S=(R/NR)^*$?
If we were asked about the original ring, I would say $\varphi(N)$ (elements disjoint to $N$ as $S=\mathbb{Z}_N^*$). However here we might have more elements? Is there a rigorous way to count them?
edit: Following the comments I have edited the question and added several more assumptions.