An nth root of unity, where n is a positive integer (i.e. n = 1, 2, 3, …), is a number z satisfying the equation:

$z^n = 1$

An nth root of unity is primitive if it is not a kth root of unity for some smaller k:

$z^k$ $\neq$ $1$ $(k = 1,2,3,...,n-1)$

Let $S(n)$ be the sum of all the primitive nth roots of unity.

Define $F(n,k) =$ $\sum_{i=1}^{n} i^k S(i)$

Enter the value of $F(10^{11},10^3)$ mod $1000000007$.