Problem about the order of a automorphic function on an elliptic curve.

Question

Let $f:X(\Gamma)\rightarrow\mathbb{C}$ be an automorphic function of weight 0, saying that $f$ is $\Gamma$-invariant.

Now at each noncusp point $\tau\in\mathbb{C}$, $f$ has an order $\nu_\tau(f)$, which is the lowest index in the Laurant expansion of $f$, and $\tau$ has a period $h_\tau$.

It is said that we must have $h_\tau\;|\;\nu_\tau(f)$.

Can anyone tell me why?