Considering our old friends the residue classes mod 44, for example, let's make a similar histogram showing what proportion of the primes show up in each one of these.
But even without thinking about sums mod 3 or anything like that, you can see this in our picture manifested as a corner with two neighbors of the same color.
A number one bigger than this was in the series expansion of a fundamental function in a totally different part of math, relevant to these things called modular forms and elliptic functions.
To take a simpler example than residue classes mod 710 think of those mod 10. Because we write numbers in base 10, this is the same thing as grouping numbers together by what their last digit is.