Eigenform
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In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form that 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. Common examples of eigenforms, and the only non-cuspidal eigenforms, are those of the Eisenstein series. Another example is the Δ function.
In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form that 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. Common examples of eigenforms, and the only non-cuspidal eigenforms, are those of the Eisenstein series. Another example is the Δ function. 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. \langle f, f \rangle = 1\, ==References==
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