HMMT 2023 A10

Let $\zeta=e^{2 \pi i / 99}$ and $\omega=e^{2 \pi i / 101}$. The polynomial

$$x^{9999}+a_{9998} x^{9998}+\cdots+a_{1} x+a_{0}$$

has roots $\zeta^{m}+\omega^{n}$ for all pairs of integers $(m, n)$ with $0 \leq m<99$ and $0 \leq n<101$. Compute $a_{9799}+a_{9800}+\cdots+a_{9998}$.