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:
where q = e2πiz. 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 of that f is not a cusp form, the eigenvalues can be given explicitly.
An eigenform which is cuspidal can be normalized with respect to its inner product:
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.
- Neal Koblitz. "III.5". Introduction to Elliptic Curves and Modular Forms.