My confusion comes from the following paper.
Ahlgren, Scott, and Matthew Boylan. "Arithmetic properties of the partition function." Inventiones mathematicae 153.3 (2003): 487-502.
In the section 5 of this paper, the authors construct \begin{align*} f_{l,j}(z):=\frac{\eta^{l^j}(l^jz)}{\eta(z)}, \end{align*} where $l$ is a prime number. And use the formula of $\eta-$quotient, they claim that \begin{align*} f_{l,j}(z)\in M_{\frac{l^j-1}{2}}\left(\Gamma_0(l^j),\left(\frac{\cdot}{l}\right)^j\right). \end{align*} I wonder why it isn't a cusp form. Furthermore, how many cusps does the congruence subgroup $\Gamma_0(l^j)$ have and how can I find all the equivalences of them?
I really need some help, and any help will be appreciated! :)