# Weight-2 modular forms under $\Gamma(N)$

I'm looking for explicit bases of weight 2 modular forms under $\Gamma(N)$, for small N (<16 would be enough). (Ideally in terms of Theta- or Eta-Functions)

It seems to me that this should probably be a problem that people have already solved but I can't find good references.

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Here's what I would do. $\Gamma(N)$ is conjugate to a subgroup of $\Gamma_1(N^2)$ via the matrix $(N,0;0,1)$, so I'd just use a computer (running e.g. SAGE of magma) to compute modular forms of weight 2, level $N^2$ and character $\chi$ for all characters of conductor dividing $N$, and then I would have figured out the answer myself. – Kevin Buzzard Nov 20 '11 at 13:24
Thanks for your reply. Unfortunately this approach does not work for me because SAGE (I don't have access to MAGMA) only spits out q-expansions, yet I need to check the transformation behavior under some modular transformations that are specifically not in $\Gamma(N)$ (e.g. S), so I need some analytical expression (maybe I was not clear enough in the original post -- sorry!) – phoboid Nov 20 '11 at 14:50
Following Kevin's suggestion of conjugating by $(N,0;0,1)$, $S$ turns into $\frac{1}{N}(0,-1;N^2,0)$ which is a constant times the Atkin-Lehner involution. If you have a newform $f$, then Atkin-Lehner will send $f$ to some constant(called a pseudo-eigenvalue) times $\bar{f}$, where the latter means "complex conjugate the fourier coefficients of $f$". The pseudo-eigenvalue can be called in SAGE by using "atkin_lehner_eigenvalue()". I'm not sure if you can just give SAGE any weight 2 $f$ of level $\Gamma(N)$, any $\gamma \in SL_2(\mathbb{Z})$, and ask for $f|\gamma$. – Barinder Banwait Nov 20 '11 at 17:51
For small $N$, this information can be extracted from Köhler's monograph Eta Products and Theta Series Identities. – S. Carnahan Apr 18 '12 at 8:13