I was looking at Convergence of the Eisenstein series this morning and am confused about something in the answer to that question.
It is written there that if we let $D=\{ z \in \mathcal{H}: |z| \geq 1, |Re(z)| \leq 1/2 \}$, $z \in D$ and $(m,n) \in \mathbb{Z}^{2}$, then $$ |mz+n|^{2} \geq m^{2}-mn+n^{2}. $$
However, this is not true if $mn<0$. E.g., $m=2$, $n=-1$ and $z=1/2+i$. Then $mz+n=2i$, $|mz+n|^{2}=4$, but $m^{2}-mn+n^{2}=7$.
Of course, I could just break up the sum defining the Eisenstein series into two subsums, one for $mn \geq 0$ and the other for $mn<0$ and then use the above argument to show that both of these sums, but neither the above answer or other textbooks I have looked in do this. They simply use the above inequality, $|mz+n|^{2} \geq m^{2}-mn+n^{2}$.
Am I missing something? Or made a glaring mistake I just can't see?