An eigenform is a modular form

Modular form

In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...

 which is an eigenvector for all Hecke operator

Hecke operator

In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations....

s T_m, m = 1, 2, 3, ….

Eigenforms fall into the realm of number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, but can be found in other areas of math and science such as analysis

Analysis

Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle , though analysis as a formal concept is a relatively recent development.The word is...

, combinatorics

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

, and physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

.

Normalization

There are two different normalizations for an eigenform (or for a modular form in general).

Algebraic normalization

An eigenform is said to be normalized when scaled so that the q-coefficient in its Fourier series

Fourier series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

 is one:


where q = e^2πiz, and a_i, i ≥ 1 turn out to be the eigenvalues of f corresponding to the Hecke operator T_i. In the case of that f is not a cusp form, the eigenvalues can be given explicitly.

Analytic normalization

As in any inner product space

Inner product space

In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

, an eigenform can be normalized with respect to its inner product:

Higher levels

In the case that the modular group

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...

 is not the full SL(2,Z), there is not a Hecke operator for each n ∈ Z, and as such the definition of an eigenform is changed accordingly: an eigenform is a modular form which is a simultaneous eigenvector for all Hecke operators that act on the space.
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