Atkin–Lehner theory is based on the concept of a newform, which 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.
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.
Then Γ0(N) is a normal subgroup of Γ0(N)+ of index 2s (where s is the number of distinct prime factors of N); the quotient group is isomorphic to (Z/2Z)s and acts on the cusp forms via the Atkin–Lehner involutions.