Modular form: Difference between revisions

Revision as of 02:21, 29 December 2022

Revision as of 02:21, 29 December 2022

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===New forms===

{{Main|Atkin–Lehner theory}}

[[Atkin–Lehner theory|New forms]] are a subspace of modular forms<ref>{{Cite web|last=Mocanu|first=Andreea|title=Atkin-Lehner Theory of <math>\Gamma_1(N)</math>-Modular Forms|url=https://andreeamocanu.github.io/atkin-lehner-theory.pdf|url-status=live|archive-url=https://web.archive.org/web/20200731204425/https://andreeamocanu.github.io/atkin-lehner-theory.pdf|archive-date=31 July 2020}}</ref> of a fixed weight <math>N</math> which cannot be constructed from modular forms of lower weights <math>M</math> dividing <math>N</math>. The other forms are called '''old forms'''. These old forms can be constructed using the following observations: if <math>M |\mid N</math> then <math>\Gamma_1(N) \subseteq \Gamma_1(M)</math> giving a reverse inclusion of modular forms <math>M_k(\Gamma_1(M)) \subseteq M_k(\Gamma_1(N))</math>.

===Cusp forms===