Eigenform

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 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. Another example is the Δ function.

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

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

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

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_i. In the case when f is not a cusp form, the eigenvalues can be given explicitly.^[1]

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

${\displaystyle \langle f,f\rangle =1\,}$

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

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.

In cybernetics, the notion of an eigenform is understood as an example of a reflexive system. It plays an important role in the work of Heinz von Foerster,^[2] and is "inextricably linked with second order cybernetics".^[3]