Atkin-Lehner Theory

Atkin-Lehner [A-L] showed how to construct in a natural way a basis for the space of modular forms of given level which are eigenfunctions for the Hecke operators prime to that level, satisfying the same formalism as for level 1. They worked on Γ₀(N). Miyake [Mi] extended this to the general case, including the modular forms in the sense of Langlands in the context of representation theory. See also Casselman [Ca]. More recently, Li [Li] reconsidered the matter in the style of Atkin-Lehner, following [A-L] very closely.