# Eigenform

An **eigenform** (meaning simultaneous Hecke eigenform with modular group SL(2,**Z**)) is a modular form which is an eigenvector for all Hecke operators *T _{m}*,

*m*= 1, 2, 3, ….

Eigenforms fall into the realm of number theory, but can be found in other areas of math and science such as analysis, combinatorics, and physics. A common example of an eigenform, and the only non-cuspidal eigenforms, are the Eisenstein series.

## 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 is one:

where *q* = *e*^{2πiz}. As the function *f* is also an eigenvector under each Hecke Operator *T _{i}*, it has a corresponding eigenvalue. More specifically

*a*

_{i},

*i*≥ 1 turns out to be the eigenvalue of

*f*corresponding to the Hecke operator

*T*. In the case of that

_{i}*f*is not a cusp form, the eigenvalues can be given explicitly.

^{[1]}

### Analytic normalization

An eigenform which is cuspidal can be normalized with respect to its inner product:

## Existence

The existence of eigenforms is a nontrivial result, but does come directly from the fact that the Hecke algebra is commutative.

## Higher levels

In the case that the modular group 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.

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}