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== Definition ==
In general,<ref>{{Cite web|last=Lan|first=Kai-Wen|title=Cohomology of Automorphic Bundles|url=http://www-users.math.umn.edu/~kwlan/articles/iccm-2016.pdf|url-status=live|archive-url=https://web.archive.org/web/20200801235440/http://www-users.math.umn.edu/~kwlan/articles/iccm-2016.pdf|archive-date=1 August 2020}}</ref> given a subgroup <math>\Gamma \subset \text{SL}_2(\mathbb{Z})</math> of [[finite index]], called an [[arithmetic group]], a '''modular form of level <math>\Gamma</math> and weight <math>k</math>''' is a [[holomorphic function]] <math>f:\mathcal{H} \to \mathbb{C}</math> from the [[upper half-plane]] such that two conditions are satisfied:
* Automorphy condition: For any <math>\gamma \in \Gamma</math> there is the equality<ref group="note">Some authors use different conventions, allowing an additional constant depending only on <math>\gamma</math>, see e.g. {{Cite web |title=DLMF: §23.15 Definitions ‣ Modular Functions ‣ Chapter 23 Weierstrass Elliptic and Modular Functions |url=https://dlmf.nist.gov/23.15#E5 |access-date=2023-07-07 |website=dlmf.nist.gov}}</ref><math>f(\gamma(z)) = (cz + d)^k f(z)</math>
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