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 N, where the levels are the nested congruence subgroups:

of the modular group, with N ordered by divisibility. That is, if M divides N, Γ₀(N) is a subgroup of Γ₀(M). The oldforms for Γ₀(N) are those modular forms f(τ) of level N of the form g(d τ) for modular forms g of level M with M a proper divisor of N, where d divides N/M. The newforms are defined as a vector subspace of the modular forms of level N, complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the Petersson inner product.

The Hecke operators, which act on the space of all cusp forms, preserve the subspace of newforms and are self-adjoint and commuting operators (with respect to the Petersson inner product) when restricted to this subspace. Therefore, the algebra of operators on newforms they generate is a finite-dimensional C*-algebra that is commutative; and by the spectral theory of such operators, there exists a basis for the space of newforms consisting of eigenforms for the full Hecke algebra.

Consider a Hall divisor e of N, which means that not only does e divide N, but also e and N/e are relatively prime (often denoted e||N). If N has s distinct prime divisors, there are 2^s Hall divisors of N; for example, if N = 360 = 2³⋅3²⋅5¹, the 8 Hall divisors of N are 1, 2³, 3², 5¹, 2³⋅3², 2³⋅5¹, 3²⋅5¹, and 2³⋅3²⋅5¹.

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