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?