Wow!

Define a Geber rotation to be the swapping of coefficient $k$ with $n+1-k$ in an $n^\text{th}$ degree polynomial . Find the number of integer rooted $2016^\text{th}$ degree polynomials such that roots are not changed when coefficients are changed in a Geber rotation and let this be $k$ . Find the remainder of $k$ when divided by 1000.