2021 Fall AMC 12A Problem 25

Let m\ge 5 be an odd integer, and let D(m) denote the number of quadruples (a_1, a_2, a_3, a_4) of distinct integers with 1\le a_i \le m for all i such that m divides a_1+a_2+a_3+a_4. There is a polynomial q(x) = c_3x^3+c_2x^2+c_1x+c_0such that D(m) = q(m) for all odd integers m\ge 5. What is c_1?

\textbf{(A)}\ {-}6\qquad\textbf{(B)}\ {-}1\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 11

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.