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My question may be basic but I can't find any answer. Let $N$ be a positive integer. I need to find the constant term (of the Fourier series) at each cusps of a modular form $$f(\tau)=\sum_{n=1}^{\infty}{a(n)q^n} \quad , \quad (q=e^{2i\pi \tau}, \tau \in \mathcal{H})$$ (where $f$ belongs to the space $\mathcal{M}_2(\Gamma_0(N))$) just by knowing its Fourier series at $i \infty$.

I know that the constant term is $0$ at $i \infty$ and since I can assume that $f$ is an eigenvalue of the Fricke involution $w_N$, I can deduce that it must also be $0$ at the cusp $0$. What else can I say ?

When $N=p$ is prime, I can conclude. But since I am working on a general result, I can't do those assumptions on $f$ : I can't explicit $f$ by a basis of $\mathcal{M}_2(\Gamma_0(N))$, neither.

Many thanks for your help !

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  • $\begingroup$ I think the key phrase you may be looking for is "Atkin-Lehner involution". $\endgroup$
    – Peter Humphries
    Commented Feb 3, 2015 at 22:57
  • $\begingroup$ If $N$ is large, then the space of such modular forms is large, and the information you've given is not sufficient to determine the constant terms at other cusps. More specifically, nothing you've said has removed the obvious ambiguity given by scalar multiplication. $\endgroup$
    – S. Carnahan ♦
    Commented Feb 4, 2015 at 0:13
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    $\begingroup$ @S.Carnahan: Certainly the Fourier coefficients at a given cusp determine the Fourier coefficients at any other cusp. I think the OP wants a simple transition formula, but I doubt such a thing exists. $\endgroup$
    – GH from MO
    Commented Feb 4, 2015 at 2:43
  • $\begingroup$ @PeterHumphries : I heard that the Atkin-Lehner involution method was only working in the case where $f$ is a cusp form and a new form. Can you please explain me the general method ? (Thanks !) $\endgroup$
    – Stabilo
    Commented Feb 4, 2015 at 7:56
  • $\begingroup$ @GHfromMO : Yes that is almost what I want : but I am only interested in the constant term. $\endgroup$
    – Stabilo
    Commented Feb 4, 2015 at 7:57

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