Multiple Levels for Modular Forms?

Question

Please pardon what I'm sure is a very naive question. I have heard discussions of modular forms for a bit and am trying to get a more fundamental understanding. People often talk about modular forms of level $N$. I have read that this means that the finite subgroup $\Gamma$ of $SL_2(\mathbb{Z})$ on which the modular form $f$ is defined contains the principal congruence subgroup $\Gamma(N)$. We can think of this subgroup as the set of matrices congruent to the identity modulo $N$. My question is this: couldn't this be true for multiple values of $N$? In other words, can a given modular form have multiple levels? Yet people seem to generally refer to a modular form as having a specific level. Is this just because we are usually interested in a specific canonical value of $N$? Thank you.

Answer 1

If $f$ is a modular form with respect to the principal congruence subgroup $\Gamma(N)$, then $f$ is also a modular form with respect to $\Gamma(\ell N)$ for any $\ell$. The converse is not true.

The natural place to talk about a modular form is the smallest level in which it can occur. There is a more technical definition, which is that the "natural" modular forms associated to a cannonical level $N$ are the "newforms" of level $N$. See this page on wikipedia for example. The LMFDB only displays information about newforms.

It might also be helpful to note that this is very similar to Dirichlet characters. Recall the mechanical definition of a mod $N$ Dirichlet character as a multiplicative function $\chi : \mathbb{Z} \longrightarrow \mathbb{C}^\times$ that satisfies $\chi(n) = \chi(n + N)$ and $\chi(n) = 0$ if $\gcd(n, N) > 1$.

For example, we have $\chi_{-4}(n) := (\frac{-4}{n})$, defined to have $\chi_{-4}(1) = 1$, $\chi_{-4}(3) = -1$, and $\chi_{-4}(n) = 0$ otherwise. Then $\chi_{-4}$ is a Dirichlet character mod $4$, but it's also a Dirichlet character mod $20$, or $444$, or any multiple of $4$. The analogous notion of "level" and "newform" with characters is "conductor" and "primitive character".