Suppose we have a bunch of coordinates $(a,b)$ such that $a+b\omega$ is an element of $\mathbb{Z}[\omega]$ where $\omega=\frac{-1+\sqrt{-7}}{2}$. Simply plotting the point $(a,b)$ in a cartesian plane is insufficient, since these elements follow a triangular lattice. What transformation would we apply to $(a,b)$ so that we could plot it in a Cartesian plane?
When $\omega=\frac{-1+\sqrt{-3}}{2}$, we have $\omega=e^{2\pi i/3}=\cos(2\pi/3)+i\sin(2\pi/3)$, so $$a+b\omega=a+b(\cos(2\pi/3)+i\sin(2\pi/3))=a+b\cos(2\pi/3)+ib\sin(2\pi/3),$$ so we apply the transformation $$(a,b)\mapsto(a+b\cos(2\pi/3), b\sin(2\pi/3))$$ and plot this in a Cartesian plane. This gives the desired triangular lattice to plot the Eisenstein integers.
However in the original case, there is no nice angle that works out as in the case of the Eisenstein integers, so what would we do here? Similarly for $11, 19, etc.?$