Highly Recommended References for the Theory of Modular Forms for Normal Subgroups of Finite Index of $PSL(2, \mathbb{Z})$

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I am looking for a highly recommended reference for the Theory of Modular Forms for Normal Subgroups of Finite Index of $PSL(2, \mathbb{Z})$.

Many references, that have been mentioned before on this website, discuss the Theory of Modular Forms for $SL(2, \mathbb{Z})$ and Congruence Subgroup of $SL(2, \mathbb{Z})$ (I don't need them at all).

All I need is a reference to understand this part of the article, starting from "1.2. Modular Forms...". Where $G=PSL(2,\mathbb{R})$. enter image description here enter image description here