2
$\begingroup$

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?

Improve this question
$\endgroup$
  • 3
    $\begingroup$ I think this related question more or less answers your question. $\endgroup$ – stupid_question_bot Aug 4 '19 at 18:03
  • 2
    $\begingroup$ 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 $q$. $\endgroup$ – Kevin Buzzard Aug 4 '19 at 20:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.