Any positive real number $x$ can be decomposed into integer and fractional parts $\lfloor x\rfloor +\{x\}$, where $\lfloor x\rfloor$ (the floor function) is an integer, and $0\le \{x\}<1$.

For positive integers $k$ and $n$, define the function $$\begin{array}{r}{f}_k(n)=\sum _{i=1}^n\left\{\frac{i^k}{n}\right\}\end{array}$$ For example, $f_5(10)=4.5$ and $f_7(1234)=616.5$.

Let $$\begin{array}{r}S(N)=\sum _{\begin{array}{c}k=1\\ k\text{ odd}\end{array}}^N\sum _{n=1}^N{f}_k(n)\end{array}$$ You are given that $S(10)=100.5$ and $S({10}^3)=123687804$.

Find $\lfloor S(33557799775533)\rfloor$. Give your answer modulo 977676779.