Let $M_0^{!}(\Gamma)$ be a space of weakly holomorphic modular functions.
Now we consider a subspace $M_0^{!,\infty}(\Gamma)$ which allows pole only at infinity.
Assume that $\Gamma=\Gamma_0(N)$ and $p$ be a prime number which divides $N$.
Let $f\in M_0^{!,\infty}(\Gamma)$. Then $f|V_p\in M_0^{!}(\Gamma_0(pN))$, where $V_p=\begin{pmatrix}p,0\\0,1 \end{pmatrix}$.
Assume that $f=q^{-m}+O(q)$ at infinity.
Obviously, $f|V_p$ has a pole at infinity with principal part $q^{-mp}$.
It looks like $f|V_p\notin M_0^{!,\infty}(\Gamma_0(pN))$.
Can we describe behavior of $f|V_P$ at other cusps?
Any idea or hint would be thankful.