For coprime positive integers $a$ and $b$, let $C_{a,b}$ be the curve defined by: $$\begin{array}{rl}x& =\cos \left(at\right)\\ y& =\cos \left(b(t-\frac{\pi }{10})\right)\end{array}$$ where $t$ varies between $0$ and $2\pi$.

For example, the images below show $C_{2,5}$ (left) and $C_{7,4}$ (right):

Define $d(a,b)=\sum (x^2+y^2)$, where the sum is over all points (x, y) at which $C_{a,b}$ crosses itself.

For example, in the case of $C_{2,5}$ illustrated above, the curve crosses itself at two points: $(0.31,0)$ and $(-0.81,0)$, rounding coordinates to two decimal places, yielding $d(2,5)=0.75$. Some other examples are $d(2,3)=4.5$, $d(7,4)=39.5$, $d(7,5)=52$, and $d(10,7)=23.25$.

Let $s(m)=\sum d(a,b)$, where this sum is over all pairs of coprime integers $a,b$ with $2\le a\le m$ and $2\le b\le m$.
You are given that $s(10)=1602.5$ and $s(100)=24256505$.

Find $s({10}^6)$. Give your answer in scientific notation rounded to $10$ significant digits; for example $s(100)$ would be given as 2.425650500e7.