Order of zero of modular form from its expansion at infinity

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1. Homework Statement


order of zero of a modular form ?

2. Homework Equations


3. The Attempt at a Solution

Apologies if this is a stupid question but I'm pretty confused.
So, a modular form ##f(t) \in M_k ## is usually given by it's expansion about ##\infty## expressed in the variable ##q=e^{2\pi i t} ## as:

## \sum\limits_{n=0}^{\infty} a_n q^n ##

where ##n## must start at zero here for ##f(t)## to be holomorphic at ##\infty## and therefore a modular form,

if instead we have:

## \sum\limits_{n=-N}^{\infty} a_n q^n ##, then the function is 'meromophic at ##\infty## ' with the order of the pole at ##\infty## being ##N##.

We use the notation ##S_k## if the constant coeffient vanishes, that is instead we have (for a modular form so holomorhphic, i.e. cusp form):

## \sum\limits_{n=1}^{\infty} a_n q^n ##


**MY QUESTION:**



how from a modular form function expressed as the fourier expansion above, can we deduce the order of a zero? The last negative coeffient gives the order of a pole, and since we set ##i\infty \to 0 ## do we not conclude that every cusp form has a zero of order infinity at infinity?

EXAMPLE 2

My notes say the following:

on the subject of ##f## a mermorphic modular form of weight ##k##:

##ord_{\infty} (f) ## = index of first non-zero coeffient in the q-expansion of ## f= ord_{q=0}(\hat{f}) ##

where ## \hat{f}(q)=f(t) ## and where the order of a point has been defined as:

##ord_p(f) = ## order of vanishing of ##f## at ##P## minus the order of the pole of ##f## at ##P##.

So, the first non-zero coefficient of the expansion, i know, gives the order of the pole, and the negative sign has also been taken into account by the wordings of my notes to take 'the index' , so this is implying that a meromorphic modular form of weight ##k## never has any zeros 'at infinity'. should this be obvious?

i dont understand.
many thanks.



thanks
 

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