2013 AMC 12A 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 \qquad \textbf{(B)} \ 401 \qquad \textbf{(C)} \ 413 \qquad \textbf{(D)} \ 431 \qquad \textbf{(E)} \ 441

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.