Eigenform

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 + q + \sum_{i=2}^\infty a_i q^i

where q = e2*πiz*. As the function f is also an eigenvector under each Hecke operator Ti, it has a corresponding eigenvalue. More specifically a**i, i ≥ 1 turns out to be the eigenvalue of f corresponding to the Hecke operator Ti. In the case when f is not a cusp form, the eigenvalues can be given explicitly.

Analytic normalization

An eigenform which is cuspidal can be normalized with respect to its inner product: :\langle f, f \rangle = 1,

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

References

1. Neal Koblitz. (1984). "Introduction to Elliptic Curves and Modular Forms".