TH-TMO 2006 Problem 5

Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$ . If $8f(0) + 9f(1) = 2006$ , compute $f(0)$ .