I wish to count all Gaussian integers (modulo $9$) that are relatively prime to $9$. So far, I can easily do this for any Gaussian prime, by drawing the fundamental parallelogram and identifying all residues within. However, I am not certain how to do this for a non-prime, namely $9$. I have observed that $9 = 3\times 3$, where $3$ is a Gaussian prime, but am not sure how to use this information to obtain the Gaussian integers that are relatively prime to $9$.
I do not need to identify these integers, only to count them.
Edit: I believe that the correct answer can be computed by counting the number of residues in the fundamental parallelogram ($81$) and subtracting the number of residues that are not coprime to $9$ (i.e., which have $3$ as a common factor, or, since $3$ is prime, which are multiples of 3.). Now, does the problem reduce to counting the possible multiples of $3$ in the fundamental parallelogram? I think I can count these by translations of a parallelogram of sidelength 3.