modular forms of half-integral weight: order, fourier series.

Question

Let $f$ be a meromorphic modular form of weight $r/2$ with $ r\in \mathbb{Z}$ for congruence subgroup $\Gamma$ and multiplicator system $v$. $f$ now has a period $N = lR$, where $R$ is the smallest translation \begin{equation} T^R = \begin{pmatrix} 1 & R \\ 0 & 1 \end{pmatrix} \in \Gamma \quad \text{and} \end{equation} $l$ is the order of the roots of unity in $v$. Then there exists the usual fourier series \begin{equation} f(z) = \sum_{n= \infty}^{\infty} a_n e^{2\pi i n z / N }. \end{equation} But if we view at the value of $v$ for the translation $v(T^R) = e^{2 \pi i \nu / l }$, we can write the fourier series as \begin{equation} f(z) e^{2 \pi i \nu / l } = \sum_{n= \infty}^{\infty} b_n e^{2\pi i n z / R} \end{equation} with $b_n := a_{\nu + l n }$. We now define the order of $f$ in $i \infty$ as \begin{equation} \mathrm{ord}(f; i \infty) = min \{ n; b_n \neq 0\}. \end{equation} My question ist: Why don't we just define the order of $f$ as $\{ n; a_n \neq 0\}$?