Why does the plot of the legendre symbol of $x^2 - y^2$ over a finite field look rectangular 
The small top-left thing is a plot of the legendre symbol of $x^2 - y^2$ over $\Bbb F_{37}$. The thing in the middle is plot for $\Bbb F_{587}$. The thing on the right is a plot of the legendre symbol of $xy$ over $\Bbb F_{587}$.
Question is, why do they ($\left(\dfrac{x^2 - y^2}{587} \right)$ and $\left(\dfrac{xy}{587} \right)$) look rectangular?
The following are plots of the same without the Legendre symbol. Notice how they don't look so rectangular.

Cheers.

I have noticed that $x^2 - y^2 = (x+y)(x-y)$ which could explain the similarity between the plots. But that's not enough to explain why the plot of $\left(\dfrac{xy}{587} \right)$ looks rectangular.
Also, the middle thing in the following picture is a plot of $x^2 + y^2$

and now we take the legendre symbols

For $x^2 + y^2$ it looks noisy and non-rectangular.

In response to Jyrki's question, the thing on the right is over $\Bbb F_{593}$

More examples of Legendre symbols of $x^2 + y^2$ where $-1$ is a quadratic residue:

Left is mod 53. Middle is mod 41. Right is mod 37.
 A: Here's what the univariate Legendre symbols $\chi(x)$ modulo 587 looks like. Along the $x$-axis there is a black square at the QRs and the rest is white.
You see that there are largely white bands and largely black bands. When you plot $\chi(xy)=\chi(x)\chi(y)$ those bands become (nearly) monochromatic regions in the 2D-plot. Because $\chi(x^2-y^2)=\chi(x-y)(x+y)$ you get the same nearly monochromatic regions but rotated by 45 degrees, because this time the factors are constant along lines with slopes $\pm1$.


Because $-1$ is a NQR modulo $587$ we cannot write the form $x^2+y^2$ as a product of two linear forms. Therefore this regions do not appear when plotting $\chi(x^2+y^2)$ modulo $587$.
A suggestion: Modulo $p=593$ we do have that $-1$ is a quadratic residue as $77^2\equiv-1\pmod{593}$. Therefore
$$x^2+y^2\equiv (x+77y)(x-77y)\pmod{593}.$$
Will those nearly monochromatic regions reapper, if you plot $\chi(x^2+y^2)$ modulo $593$? That $77$ may be too awkward a slope to show up :-)


When you leave out the Legendre symbol, the changes become more gradual. That may be all you need to remove those rectangles.
