A big sum 2

Let $$S(m,n)=\sum _{k=1}^m\sum _{x=1}^n(x^k\cdot \lfloor \frac{\pi x}{k}{\rfloor }^k)$$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.

You are given $S(1,10)=1182$, $S(3,100)=16716537706541$ and $S(10,{10}^4)\mathrm{mod}1000000007=211872856$.

Find $S(20,{10}^{20})\mathrm{mod}1000000007$.

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Hint: solve Problem 399 - A Big Sum first and there are some clues in the secret forum.