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I am just a beginner in modular forms.. It seems for me that lots of the work has been done for the cases of spaces of modular forms for $\Gamma_0(N)$ or $\Gamma_1(N)$ with some Dirichlet character $\chi_d$.

I am curious about the cases $\Gamma^0(N)$ and $\Gamma^1(N)$ with some Dirichlet character $\chi_d$.

Here, $\Gamma^0(N)$ is the congruence subgroup of $SL_2(\mathbb{Z})$, consists of $A=(a_{ij})$, where the upper right corner $a_{12}\cong 0$ mod $N$.

Could anyone give me some reference or some idea about how to get space of modular forms of $\Gamma^0(N)$ and $\Gamma^1(N)$ with Dirichlet character $\chi_d$? Or is it possible to get them from the cases of $\Gamma_0$ and $\Gamma_1$?

Thank you so much

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5  
What are $\Gamma^0$ and $\Gamma^1$? (They appear sometimes in papers of mine but just as typos for $\Gamma_0$ or $\Gamma_1$) –  Joël Jan 21 '12 at 17:50
    
I so sorry for the confusion. I added the definition of $\Gamma^0$ –  YOURS Jan 21 '12 at 18:57
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$\Gamma^0$ and $\Gamma^1$ are equivalent to $\Gamma_0$ and $\Gamma_1$ via conjugation by $S = ({\phantom-0\phantom.1\atop-1\phantom.0})$. So the spaces of modular forms are equiavalent with those of $\Gamma_0$ and $\Gamma_1$ via composition with $S$. –  Noam D. Elkies Jan 21 '12 at 21:03
    
What I am looking at is one particular group $\Gamma^0(N)$ with Dirichlet character $\chi_d$. Is the space of modular forms of $\Gamma_0(N)$ with $\chi_d$ the same as the space of modular forms of $\Gamma^0(N)$ with $\chi_d$? –  YOURS Jan 21 '12 at 22:01
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Conjugation by $S$ takes the $2 \times 2$ matrix $(a,b;c,d)$ to $(-d,c;b,-a)$ or equivalently $(d,-c;-b,a)$. Since $ad\equiv 1 \bmod N$ on $\Gamma_0(N)$ (and on $\Gamma^0(N)$), the action of $S$ takes forms with character $\chi$ to forms with character $1/\chi = \overline\chi$. –  Noam D. Elkies Jan 21 '12 at 22:52
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