Sinful sum

Algebra Level 5

Let NN be the number of ordered quadruples (a,b,c,d)(a,b,c,d) of integers, each of which are from 10-10 to 1010 (inclusive), such that {a+b+c+d=0sina+sinb+sinc+sind=0.\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?N?

Details and assumptions

The calculations are done in radians, not degrees.

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