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Let's use $M_k^{!}(\Gamma_{0}(N))$ to denote weakly holomorphic modular form with weight k, level $\Gamma_0(N)$(in particular, k might be negative). Then obviously $M_0^{!}(\Gamma_{0}(N))$ acts on it by multiplication since the weight will not be changed. Actually we know $j$ invariant is a generator for $M_0^{!}(\Gamma_{0}(N))$.

My question: Is $M_k^{!}(\Gamma_{0}(N))$ a finitely generated as a $M_0^{!}(\Gamma_{0}(N))$-module? If it is, can we find some generator?

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For $N=1$ this module is cyclic: see the first two pages of Duke-Jenkins (2008), which might also help to find the generators for $N>1$.

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    $\begingroup$ Here $M_{k}^{!}(\Gamma_{0}(N))$ is the space of weight $k$ forms that are holomorphic on the upper half plane, but can have poles (of arbitrary order) at the cusps, so it is not finite-dimensional over $\mathbb{C}$. $\endgroup$
    – Jeremy Rouse
    Commented Nov 25, 2019 at 1:09
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    $\begingroup$ Thanks for the reply! Actually I am considering weakly holomorphic modular form, not just cusp form. And in particular I am more interested in negative weight case, and this case is more mysterious I guess? $\endgroup$
    – user330928
    Commented Nov 25, 2019 at 2:25
  • $\begingroup$ @JeremyRouse: You are right, I overlooked this. Anyways, I am keeping the second half of my answer as it was correct. $\endgroup$
    – GH from MO
    Commented Nov 25, 2019 at 3:47
  • $\begingroup$ @user330928: I agree. Apologies for being careless. $\endgroup$
    – GH from MO
    Commented Nov 25, 2019 at 3:49

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