# order of zero of modular form from it's expansion at infinity

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,

$\sum\limits_{n=-N}^{\infty} a_n q^n$, 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.