- Atkin-Lehner theory
In

mathematics , the**Atkin-Lehner theory**is an algebraic part of the theory ofmodular form s, in which the concept of**newform**is defined. A newform is acusp form 'new' at a given "level" "N", where the levels refer to the nested subgroups:Γ("N")

of the

modular group , with "N" ordered bydivisibility . That is, if "M" divides "N", we have that Γ("N") is asubgroup of Γ("M"). The "oldforms" for Γ("N") are those modular forms of level "N" that already have level "M" with "M" a proper divisor of "N". The newforms are defined as a vector subspace of the modular forms of level "N", complementary to the space spanned by the oldforms.The main question is to relate and reconcile the definition of newform to the action of the

Hecke operator s. Theseself-adjoint operators, which act on the space of allcusp form s happen to preserve the space of newforms. Thus, there exists a basis for the space of newforms consisting of eigen forms for the full Hecke algebra. A simple analogue in the theory ofDirichlet character s is that newforms play the role of "primitive" Dirichlet characters. When it comes to constructingL-function s, non-primitive Dirichlet characters may have some 'missing'Euler factor s. Analogously, only newforms that are eigenforms of the Hecke algebra can be expected to present correctly-formed associatedDirichlet series .**References*** A. Atkin and J. Lehner. "Hecke operators on Γ

_{0}("m")". Math. Ann. , (185):134--160, 1970.

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**Atkin–Lehner theory**— In mathematics, Atkin–Lehner theory is part of the theory of modular forms, in which the concept of newform is defined in such a way that the theory of Hecke operators can be extended to higher level. A newform is a cusp form new at a given level … Wikipedia**A. O. L. Atkin**— Arthur Oliver Lonsdale Atkin (July 31, 1925 – December 28, 2008), who published under the name A. O. L. Atkin, was a Professor Emeritus of mathematics at the University of Illinois at Chicago. As an undergraduate during World War II, he worked at … Wikipedia**A. O. L. Atkin**— Arthur Oliver Lonsdale Atkin, zitiert als A. O. L. Atkin, er selbst benutzte den Vornamen Oliver, (* 31. Juli 1925; † 28. Dezember 2008 in Maywood (Illinois)) war ein britisch US amerikanischer Mathematiker, der sich mit Zahlentheorie und… … Deutsch Wikipedia**List of mathematics articles (A)**— NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A … Wikipedia