$(x,y)$ is called a nested radical pair if $x$ and $y$ are non-zero integers such that ${\displaystyle \frac{x}{y}}$ is not a cube of a rational number, and there exist integers $a$, $b$ and $c$ such that:

$$\sqrt{\sqrt[3]{x}+\sqrt[3]{y}}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$$

For example, both $(-4,125)$ and $(5,5324)$ are nested radical pairs:

$$\begin{array}{r}\begin{array}{rl}\sqrt{\sqrt[3]{-4}+\sqrt[3]{125}}& =\sqrt[3]{-1}+\sqrt[3]{2}+\sqrt[3]{4}\\ \sqrt{\sqrt[3]{5}+\sqrt[3]{5324}}& =\sqrt[3]{-2}+\sqrt[3]{20}+\sqrt[3]{25}\end{array}\end{array}$$

Let $H(N)$ be the sum of $|x|+|y|$ for all the nested radical pairs $(x,y)$ where $|x|\le |y|\le N$.
For example, $H({10}^3)=2535$.

Find $H({10}^{15})$. Give your answer modulo ${1031}^3+2$.