Eigenform

In mathematics, aneigenform(meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is amodular formwhich is aneigenvectorfor allHecke operatorsT_m,m= 1, 2, 3,....

Eigenforms fall into the realm ofnumber theory,but can be found in other areas of math and science such asanalysis,combinatorics,andphysics.A common example of an eigenform, and the only non-cuspidal eigenforms, are theEisenstein series.Another example is theΔ function.

Normalization

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

Algebraic normalization

An eigenform is said to benormalizedwhen scaled so that theq-coefficient in itsFourier seriesis one:

$f=a_{0}+q+\sum _{i=2}^{\infty }a_{i}q^{i}$

whereq=e^2πiz.As the functionfis also an eigenvector under each Hecke operatorT_i,it has a corresponding eigenvalue. More specificallya_i,i≥ 1 turns out to be the eigenvalue offcorresponding to the Hecke operatorT_i.In the case whenfis not a cusp form, the eigenvalues can be given explicitly.

Analytic normalization

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

$\langle f,f\rangle =1\,$

Existence

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

Higher levels

In the case that themodular groupis not the full SL(2,Z), there is not a Hecke operator for eachn∈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.

In cybernetics

Incybernetics,the notion of an eigenform is understood as an example of a reflexive system. It plays an important role in the work ofHeinz von Foerster,and is "inextricably linked withsecond order cybernetics".