Imagery from double character sums

Things on the graphs:

For the character group of size n we plot the real part against the imaginary part of exp( 2*pi*1i*A*k/n ) where A is the least residue set modulo n with elements relativily prime to n.
We do this for every character k in the character group, k=0 is at the 0 radians point; and then we go round counter clockwise, arriving at n graphs layed out around the circle.
If the sum of these points for a fixed value of k, sum( exp( 2*pi*1i*A*k/n ) ), is zero then the point set is balanced and we plot the points in blue; otherwise the points are plotted in red.

Things to notice:

Square free numbers will only have unbalanced configurations (colored red).
The starting point (the full colored dot) is the regular progression around the circle.
Pure powers have their base unbalanced (the red parts are exactly the figure of the base on its own) and all the rest balanced (colored blue).
A square free product of primes will obviously give that many unbalanced configurations.
A non square free product of primes will have the square free proportion as unbalanced configurations.
The small circles are always symmetric around the x-axis; which means the character sum is always a real number.
The big circle is also symmetric around the x-axis.
If one of the factors is 2 then the big circle is also symmetric around the y-axis.
If we have n = 2² p, p an odd prime, then we have alternating colors.
More general, for n = p^a q^b we have alternations of a red one and p^a-1q^b-1-1 blue ones.