let $f:C\longrightarrow C$ and $f(z)=z^2+zi+1$,find the number of $z$,such $Im(z)>0$
and $|Im(f(z))|\le 10,|Re(f(z))|\le 10$,and $ Im(f(z)) ,Re(f(z)) $are integer numbers.
where $i=\sqrt{-1}$,

$A:399, B:401, C:413, D:431, E:441$

my idea:let $z=a+bi,b>0$, then 
$$f(z)=z^2+iz+1=(a+bi)^2+i(a+bi)+1=(a^2-b^2-b+1)+(2ab+a)i$$

so:
$|a^2-b^2-b+1|\le 10,|2ab+a|\le 10$,and
$a^2-b^2-b+1\in Z,2ab+a\in Z,b>0$
then this problem equalitly find the number pairs$(a,b)$? But I can't find .

Thank you everyone,