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 $10$?
A. $\, 399 \quad$ B. $\, 401 \quad$ C. $\, 413 \quad$ D. $\, 431 \quad$
I cannot make much progress here. I tried putting $z=a+ib$. I also tried putting $z=e^{i\theta}$. But, none of them worked. Any kind of help would be appreciated.