In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form which is an eigenvector for all Hecke operators Tm, 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.
As the function f is also an eigenvector under each Hecke operator Ti, it has a corresponding eigenvalue.
More specifically ai, i ≥ 1 turns out to be the eigenvalue of f corresponding to the Hecke operator Ti.
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.
It plays an important role in the work of Heinz von Foerster,[2] and is "inextricably linked with second order cybernetics".