If $\Gamma \le SL(2;\mathbb{Z})$ is a subgroup for which we know generators, is there a principle for constructing modular forms for $\Gamma$ out of those for $SL(2;\mathbb{Z})$?
An example of what I want to do is this. If $\Gamma$ is generated by $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},$ then the modular forms of weight $4$ are spanned by $E_4(z)$ and $E_4(\frac{z+1}{2}).$ I would like to know the intuition for plugging $\frac{z+1}{2}$ into $E_4$.
