of the modular group, with N ordered by divisibility. That is, if M divides N, we have that Γ(N) is a subgroup 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 operators. These self-adjoint operators, which act on the space of all cusp forms 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 of Dirichlet characters is that newforms play the role of primitive Dirichlet characters. When it comes to constructing L-functions, non-primitive Dirichlet characters may have some 'missing' Euler factors. Analogously, only newforms that are eigenforms of the Hecke algebra can be expected to present correctly-formed associated Dirichlet series.