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:
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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.
If $M$ divides $N$, and $d$ divides $N/M$, then for any modular form $f(\tau)$ on $\Gamma_0(M)$, the function $f(d\tau)$ is a modular form on $\Gamma_0(N)$.
Let's write $V_N$ for the vector space of cuspforms on $\Gamma_0(N)$ (of some fixed weight). Then the above map, i.e. $f(\tau) \mapsto f(d\tau)$, gives an embedding of $V_M$ into $V_N$; write $V_{M,d}$ its image.
Then the space of old forms in $V$ is the subspace of $V$ obtained by taking the sum over all proper divisors $M$ of $N$ and all possible corresponding choices of $d$ of the subspaces $V_{M,d}$. We use the word old for them because these forms, while they have level $N$, actually come from a smaller level (namely $M$), and so are "old" from the point of view of the level $N$. The space of new forms is the orthogonal complement in $V$ of the space of old forms; elements of $V$ are genuinely new to level $N$, hence their name.