I apologize in advance for what is probably a very naive question: I'd like to understand the Fourier coefficients of newforms, and so I was wondering what exactly was known about them (I do know that the situation isn't as straightforward as for Eisenstein series). I have looked at the algorithms in Modular Forms a Computational Approach, but I was hoping for more explicit expressions.
In particular, when I run the command Newforms(CuspForms(N,k)) for lowish weight and level in Magma, the q-expansions that are outputted usually look "nice" (for example one q-expansion will differ from another by a quadratic character). I was interested in more information on this, as well as any explanation for why Magma outputs the expansions in the form that they do. Thanks!