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"!