I wish to find the addition and multiplication tables of Eisenstein integers modulo $(1-\omega)^2 = -3\omega$. In drawing the fundamental parallelogram with vertices at $0, z = - 3\omega, z\omega = 3+3\omega$ and $(1+\omega)z = 3$, I discovered that the set of Eisenstein integers inside this parallelogram is $\{0, 1, 2, 1+\omega, 2 + \omega, 2 + 2\omega, -\omega, 1-\omega, -2 \omega\}$. Now I am to find its addition and multiplication tables which are 9x9 but I am not certain how to compute this. For instance, to what is $3 mod (-3\omega)$ congruent? I can use the fundamental parallelogram, and if I am correct, I think this point "translates" back to $0$? Can this be verified algebraically?
I am told that the multiplication tables is the "same" as those for the rational integers modulo $9$, but I cannot make the identification as to what numbers in the set $\{0, 1, 2, 1+\omega, 2 + \omega, 2 + 2\omega, -\omega, 1-\omega, -2 \omega\}$ correspond to $\{0, 1, 2, 3, 4, 5, 6, 7, 8\}$.