TITLE: What is the idea behind the new/old form theory of modular forms
QUESTION [5 upvotes]: I am interested in the theory of modular forms, since they have nice transformation laws and a connection to arithmetic using the Euler product which one can form using the theory of Hecke operators.
The vector space of modular forms with respect $\Gamma_0(N)$ can be decompose to so called a new and old forms.
My question: 

what is the idea behind this decomposition and why it is useful?
why they called "new" and "old"?

Thanks

REPLY [2 votes]: In short, some forms on $\Gamma_0(N)$ are also forms for $\Gamma_0(M)$ for $M < N$. They don't fit all the patterns as nicely, so we call forms that aren't in $\Gamma_0(M)$ for $M < N$ a special name: "newforms."
This isn't so new. For example, when looking at cyclotomic polynomials, all the $n$th roots of unity aren't the same. For example, $e^{2\pi i/3}$ is a third root of unity and a sixth root of unity (and 12th... and on and on). But do we really think of it as a sixth root of unity? We call it a primitive third root of unity, and the primitive sixth roots of unity are just the sixth roots of unity that are not an $M$th root of unity for some $M < 6$. It's the same idea.