Define the $(N,M)$-functional inverse of $x^2$ to be the lowest-degree monic polynomial $Q(x)$ such that $Q(n^2)\equiv n(\mathrm{mod}M)$ for all integers $0\le n\le N$ and all coefficients are non-negative and smaller than $M$.

For example, the $(2,7)$-functional inverse of $x^2$ is $x^3+3x^2+4x$.

Find the coefficient of $x^{10}$ in the $({10}^7,{10}^9+7)$-functional inverse of $x^2$.