Eccoci.

Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that is coprime with the current size of the pile. Two players remove stones in turn. The player who removes the last stone wins.

Let $L(n, k)$ be the number of losing starting positions for the first player, assuming perfect play, when the game is played with $k$ piles, each having between $1$ and $n - 1$ stones inclusively.

For example, $L(5, 2) = 6$ since the losing initial positions are $(1, 1)$, $(2, 2)$, $(2, 4)$, $(3, 3)$, $(4, 2)$ and $(4, 4)$.
You are also given $L(10, 5) = 9964$, $L(10, 10) = 472400303$, $L(10^3, 10^3) \bmod 1\,000\,000\,007 = 954021836$.

Find $L(10^7, 10^7)\bmod 1\,000\,000\,007$.