AMC 12 2013a Problem 25

Let $f : \mathbb{C} \to \mathbb{C}$ be defined by $f(z) = z^2 + iz + 1$. How many complex numbers $z$ are there such that $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $10$?

\textbf{(A)} \ 399 \textbf{(B)} \ 401 \textbf{(C)} \ 413 \textbf{(D)} \ 431 \textbf{(E)} \ 441