# Dirichlet series associated with holomorphic cusp forms

Level one modular forms. A cusp form of weight for the full modular group is a holomorphic function on the upper half-plane which satisfies

for all integers with and also has the property that . Cusp forms for the whole modular group exist only for even integers and . The cusp forms of a given weight of this form make a complex vector space of dimension if and of dimension if . Each such vector space has a special basis of Hecke eigenforms which consist of functions for which

The Fourier coefficients are real algebraic integers of degree equal to the dimension of the vector space . Thus, when the spaces are one dimensional and the coefficients are ordinary integers. The L-function associated with a Hecke form of weight is given by

By Deligne's theorem for a real . It is conjectured (Sato-Tate) that for each the is uniformly distributed on with respect to the measure . We write where ; then

The functional equation satisfied by is

Higher level forms. Let denote the group of matrices with integers satisfying and . This group is called the Hecke congruence group. A function holomorphic on the upper half plane satisfying

for all matrices in and is called a cusp form for ; the space of these is a finite dimensional vector space . The space above is the same as . Again, these spaces are empty unless is an even integer. If is an even integer, then

where is the index of the subgroup in the full modular group :

is the number of cusps of :

is the number of inequivalent elliptic points of order 2:

and is the number of inequivalent elliptic points of order 3:

It is clear from this formula that the dimension of grows approximately linearly with and .

For the spaces the issue of primitive forms and imprimitive forms arise, much as the situation with characters. In fact, one should think of the Fourier coefficients of cusp forms as being a generalization of characters. They are not periodic, but they act as harmonic detectors, much as characters do, through their orthogonality relations (below). Imprimitive cusp forms arise in two ways. Firstly , if , then for any integer Secondly, if , then for any . The dimension of the subspace of primitive forms is given by

where is the multiplicative function defined for prime powers by if , if , and if . The subspace of newforms has a Hecke basis consisting of primitive forms, or newforms, or Hecke forms. These can be identified as those which have a Fourier series

where the have the property that the associated L-function has an Euler product

The functional equation satisfied by is

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