Editing Modular form (section)

==Types==

===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 level <math>N</math> which cannot be constructed from modular forms of lower levels <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===
{{Main|Cusp form}}
A [[cusp form]] is a modular form with a zero constant coefficient in its Fourier series. It is called a cusp form because the form vanishes at all cusps.