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In the definition of a (weakly holomorphic) modular form we require a specific growth behaviour at all the cusps. I assume that this requirement is not void, i.e. not automatically satisfied.

However, I have never seen an example of a modular form, which satisfies the growth condition at one cusp but not at another.

So my question is: Can somebody give an example of a function which transforms like a modular form (I do not care about the weight or the congruence subgroup) and which is weakly holomorphic (i.e. has at most a pole) a some cusp but not at some other cusp (i.e. has an essential singularity).

Are there also examples which are interesting from the point of arithmetic geometry/number theory/...

EDIT: Does somebody also know an example where the weight is non-zero

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Modular form with pole of infinite order

In the definition of a (weakly holomorphic) modular form we require a specific growth behaviour at all the cusps. I assume that this requirement is not void, i.e. not automatically satisfied.

However, I have never seen an example of a modular form, which satisfies the growth condition at one cusp but not at another.

So my question is: Can somebody give an example of a function which transforms like a modular form (I do not care about the weight or the congruence subgroup) and which is weakly holomorphic (i.e. has at most a pole) a some cusp but not at some other cusp (i.e. has an essential singularity).

Are there also examples which are interesting from the point of arithmetic geometry/number theory/...