Eigenform

An eigenform (meaning simultaneous Hecke eigenform) 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.

Contents

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:

f = a_0 %2B q %2B \sum_{i=2}^\infty a_i q^i

where q = e2πiz, and ai, i ≥ 1 turn out to be the eigenvalues of f corresponding to the Hecke operator Ti. In the case of that f is not a cusp form, the eigenvalues can be given explicitly.[1]

Analytic normalization

As in any inner product space, an eigenform can be normalized with respect to its inner product:

\langle f, f \rangle = 1\,

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.

  1. ^ Neal Koblitz. "III.5". Introduction to Elliptic Curves and Modular Forms.