The following image is a much larger
version of the table on the wikipedia page for the Legendre Symbol. I sort of like throwing big
data at problems like this, or, to express it with greater clarity: the table
on the wikipedia page is too small! source code tarball. Quadratic nonresidues are in red. Quadratic residues
are in blue (1) or green (0). I wonder if the radial striping also appears
in the Hilbert and Artin symbols? Probably could use the attention of a
number theorist proper.

Bounty: I am offering a bottle of whisky to anyone who can successfully translate Jacob Lurie's Higher Topos Theory (and/or the stacks project) into the analytic combinatorical language of the manner of Flajolet's Analytic Combinatorics.