# Modular forms that are not $\Gamma_1(N)$

So I had to study some modular forms theory for a research program and I don't understand something. Given a positive integer $$N$$, we can consider 3 classes of holomorphic modular forms (over $$\mathbb{Q}$$): $$\Gamma_0(N)$$-modular forms, $$\Gamma_1(N)$$-modular forms, $$\Gamma(N)$$-modular forms. For $$N=1$$ they coincide, in general the first is contained in the second and the second is contained in the third.

For some reason I don't see much discussion of $$\Gamma(N)$$-modular forms that are not $$\Gamma_1(N)$$. Is it because they are too boring or because any interesting information about them can be somehow recovered from information about $$\Gamma_1(N)$$-modular forms?

As a side note LMFDB seemingly defines modular forms of level $$N$$ as $$\Gamma(N)$$-modular forms and it is possible to require newforms to have certain Dirichlet character in the queries. I am not aware of any way to associate a Dirichlet character to a modular form that is not $$\Gamma_1(N)$$ (they are not allowing non-trivial Nebentypus as far as I can see). Is it because LMFDB ignore non-$$\Gamma_1(N)$$-modular forms?

• I think this related question more or less answers your question. – stupid_question_bot Aug 4 '19 at 18:03
• If $f(q)$ has level $\Gamma(N)$ then it can be written as a power series in $q^{1/N}$, and the form $f(q^N)$ is a modular form of level $\Gamma_1(N^2)$. The group $\Gamma_1(M)$ is more convenient because it contains the matrix $(1,1;0,1)$ so you don't have to keep faffing around with fractional powers of $q$. – Kevin Buzzard Aug 4 '19 at 20:32