A matrix $M$ is called idempotent if $M^2=M$.
Let $M$ be a three by three matrix : $M=\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$.
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 $C(200)$.