Fourier coefficients of newforms 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!
 A: I'm not quite sure what kind of information you're expecting, but there is a basis of the space of newforms which consists of eigenfunctions of the Hecke operators, which means there is an Euler product expression ; this is Atkin-Lehner theory. 
One could also mention growth conditions...
Serre's "Cours d'arithmétique" is a nice reference, and there are english translations.
EDIT. Yes, in the reference I gave, Serre limits himself to level one -- but he does cover Hecke operators and their eigenfunctions and discuss coefficients growth and Euler product, so the basics are nicely laid out.
Other references : Miyake's "Modular forms", Hida's "Elementary theory of $L$-functions and Eisenstein series", Bump's "Automorphic forms and representations"... and of course, there's Shimura's "Introduction to the arithmetic theory of automorphic functions"!
EDIT2. Manin&Panchishkin's "Introduction to modern number theory" has a nice exposition of the Atkin-Lehner theory too. I can't help but notice that I'm still not quite sure what type of information was expected... and mostly keep on piling on references...
A: You should have a look at Ken Ono's nice book: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series. Theorem 2.27 in this book is a theorem of Atkin and Lehner that "captures the essential properties of a newform" in the integer weight case. Read section 3.4 for Kohnen's theory for the half-integer weight case.  
