I'am reading "A First Course in Modular Forms". On page 74, the author define the order of an automophic form of weight k at $\infty$ to be half of the order in the expansion of $q^\prime=e^{2\pi iz/2h}$, where h is the period of $\infty$. But when $\Gamma_{\infty}=\left\langle-\left[\begin{array}{ll}1 & h \\ 0 & 1\end{array}\right]\right\rangle$, at the beginning of this section, $\nu_{\infty}(f)$ should be computed using the expansion in the form $q^\prime=e^{2\pi iz/2h}$, hence we have $\nu_{\pi(\infty)}(f)=\frac{\nu_{\infty}(f)}{2}$.
I'm confused about the arthors say that when k is even, we have $\nu_{\pi(\infty)}(f)=\nu_{\infty}(f)$. Do the arthors just define $\nu_{\pi(\infty)}(f)$ to be $\nu_{\infty}(f)$ in this case?