Idempotent Matrices
A matrix $M$ is called idempotent if $M^2 = M$.
Let $M$ be a three by three matrix : $M=\begin{pmatrix} a & b & c\\ d & e & f\\ g &h &i\\ \end{pmatrix}$.
Let $C(n)$ be the number of idempotent three by three matrices $M$ with integer elements such that
$ -n \le a,b,c,d,e,f,g,h,i \le n$.
$C(1)=164$ and $C(2)=848$.
Find $C(200)$.