# Sinful sum

**Algebra**Level 5

Let \(N\) be the number of ordered quadruples \((a,b,c,d)\) of integers, each of which are from \(-10\) to \(10\) (inclusive), such that \[\begin{cases} a+b+c+d & =0\\ \sin a + \sin b+ \sin c +\sin d &=0. \end{cases}\] What are the last three digits of \(N?\)

**Details and assumptions**

The calculations are done in radians, not degrees.