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"!

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...

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.

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"!