We define ${\nu }_2(n)$ to be the largest integer $r$ such that $2^r$ divides $n$. For example, ${\nu }_2(24)=3$.

Define ${\displaystyle S(n)=\sum _{k=1}^n(-2{)}^k(\genfrac{}{}{0ex}{}{2k}{k})}$ and $u(n)={\nu }_2(3S(n)+4)$.

For example, when $n=4$ then $S(4)=980$ and $3S(4)+4=2944=2^7\cdot 23$, hence $u(4)=7$.
You are also given $u(20)=24$.

Also define ${\displaystyle U(N)=\sum _{n=1}^{N}u(n^3)}$. You are given $U(5)=241$.

Find $U({10}^4)$.