Recall the blancmange function from Problem 226: $T(x)=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{s(2^{n}x)}{2^n}}$, where $s(x)$ is the distance from $x$ to the nearest integer.

For positive integers $k,t,r$, we write $$F(k,t,r)=(2^{2k}-1)T\left(\frac{(2^t+1{)}^r}{2^k+1}\right).$$ It can be shown that $F(k,t,r)$ is always an integer.
For example, $F(3,1,1)=42$, $F(13,3,3)=23093880$ and $F(103,13,6)\equiv 878922518(\mathrm{mod}1000062031)$.

Find $F({10}^{18}+31,{10}^{14}+31,62)$. Give your answer modulo $1000062031$.