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Unfortunately I am missing the definition of $\mathbb{M}_k^!$, where $\mathbb{M}_k$ is the linear space of modular forms of weight $k$ to $SL_2(\mathbb{Z})$. To give you some context, what it could mean, there is the following task.

For $f \in \mathbb{M}_k^!$, we set $d(f)=\frac{1}{2\pi i}f'-\frac{k}{4\pi^2}G_2f$, where $G_2$ is a Eisenstein series. The Claim is $$d(f)\in \mathbb{M}_{k+2}^!.$$ I hope someone is familiar with this notation and even got a little hint. Thanks

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    I've seen this notation used to refer to weakly holomorphic modular forms, which are allowed to have poles at cusps. – 2017-01-28
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    @carmichael561, So for example $j \in \mathbb{M}_0^!$? Can you maybe give me another example of a weakly holomorphic modular form? And do you know if the Claim is true if it is referring to this definition? Thanks a lot :) – 2017-01-28
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    I think $d(f)$ is even in $M_{k+2}$, but I haven't thought about this in a while. – 2017-01-28
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    @carmichael561, Do you know what the purpose of the Eisenstein Series $G_2$ is and why it is used? Thanks – 2017-01-28
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    Do you know how $G_2$ transforms under the modular group? It's not quite a modular form, but neither is $f^{\prime}$, so I believe that $d(f)$ is constructed so that the "extra" terms cancel. – 2017-01-28
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    @carmichael561. Yes I have the transformations. That's probably the right approach. I will try it. The only thing which was confusing me was the "!". There should not be reason why it should not work for usual holomorphic modular forms right? – 2017-01-28
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    Perhaps the problem is just trying to avoid checking whether $d(f)$ is holomorphic at the cusp. – 2017-01-28
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    [Page 29](http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/114.pdf) of here gives a definition. I haven't read it all in detail (I studied only the harmonic maass forms), but maybe the earlier pages could give insight? It's the same definition as @carmichael561 used, but with something called a nebentypus. I'm sorry for not remembering more of this course than the notation. – 2017-03-16
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    @mdave16 thanks for the reference. As far as i remember, the ! stands for the modular forms which are holomorphic on H but can have pole at $\infty $ – 2017-03-16

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