A modular form is holomorphic at $\infty$ means

Question

I have trouble understanding the definition of the modular forms for the congruence subgroup.
In my text book, the condition "holomorphic at all cusps" is described by :
the Fourier coefficient of $f \vert _k \gamma =\sum a_n q^n$ is $0$ if $n<0.$
For a congruence subgroup $\Gamma$ of level $N$, since it contains $\begin{pmatrix}1&N\\0&1\end{pmatrix}$, every modular form $f$ for $\Gamma$ has a period $N$ and hence a Fourier expansion $\sum a_n q_{N}^{n}$ where $q_{N}=e^{2\pi i z/N}.$
My Question is :
Why is the $q$-expansion at $i \infty$ of $f$ is of the form $\sum a_n q^n$? I think that it has to be of the form $\sum a_n q_{N}^{n}$ since it has period $N.$