Hurray! They're integer roots.

Let $B$ be the set of all integers. Given $2$ polynomials

$p(x)=x^3+ax^2+bx+c$

with $a,b,c$ are elements of $B$ and

$q(x)=x^2-40x+2410$

Assuming that $p(x)=0$ has $3$ distinct integer roots, $p(2005)=-2005$ and $p(q(x))=0$ doesn't have any real roots, what are the last $3$ digits of the absolute value of the sum of all possible values of $a$ ?