A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:

$$x(t)=(R-r)\cos (t)+r\cos (\frac{R-r}{r}t)$$ $$y(t)=(R-r)\sin (t)-r\sin (\frac{R-r}{r}t)$$

Where $R$ is the radius of the large circle and $r$ the radius of the small circle.

Let $C(R,r)$ be the set of distinct points with integer coordinates on the hypocycloid with radius R and r and for which there is a corresponding value of t such that $\sin (t)$ and $\cos (t)$ are rational numbers.

Let $S(R,r)=\sum _{(x,y)\in C(R,r)}|x|+|y|$ be the sum of the absolute values of the $x$ and $y$ coordinates of the points in $C(R,r)$.

Let $T(N)=\sum _{R=3}^N\sum _{r=1}^{\lfloor \frac{R-1}{2}\rfloor }S(R,r)$ be the sum of $S(R,r)$ for R and r positive integers, $R\le N$ and $2r<R$.

You are given:

$C(3,1)$	=	$\{(3,0),(-1,2),(-1,0),(-1,-2)\}$
$C(2500,1000)$	=	$\{(2500,0),(772,2376),(772,-2376),(516,1792),(516,-1792),(500,0),(68,504),(68,-504),$
		$(-1356,1088),(-1356,-1088),(-1500,1000),(-1500,-1000)\}$

Note: $(-625,0)$ is not an element of $C(2500,1000)$ because $\sin (t)$ is not a rational number for the corresponding values of $t$.

$S(3,1)=(|3|+|0|)+(|-1|+|2|)+(|-1|+|0|)+(|-1|+|-2|)=10$

$T(3)=10;T(10)=524;T(100)=580442;T({10}^3)=583108600$.

Find $T({10}^6)$.