I have been reading the book "Prime of the form x^2 + ny^2" by David Cox as part of my reading project, and I wanted to understand the concept of genus better. I referred to Buell's book on b.q.fs and understood some of it, but I have the following questions:
I haven't come across the generic characters that Buell defined anywhere else. All other sources talk about a homomorphism from integers to $\pm 1$, but they do not go into more detail about $\chi$. Why is that?
I looked at the genera classification for the example D = -1520. How exactly do we classify them using those characters? Do we simply look at an r that is represented by a certain reduced form, and then look at the character value assigned to it?