A family of operators mapping each space of modular forms onto itself. For a fixed integer k and any positive integer n, the Hecke operator is defined on the set of entire modular forms of weight k by

For n a prime p, the operator collapses to

If has the Fourier series

then has Fourier series

where

(Apostol 1997, p. 121).

If , the Hecke operators obey the composition property

Any two Hecke operators T(n) and T(m) on commute with each other, and moreover

(Apostol 1997, pp. 126-127).

Each Hecke operator has eigenforms when the dimension of is 1, so for k = 4, 6, 8, 10, and 14, the eigenforms are the Eisenstein series , , , , and , respectively. Similarly, each has eigenforms when the dimension of the set of cusp forms is 1, so for k = 12, 16, 18, 20, 22, and 26, the eigenforms are , , , , , and , respectively, where is the modular discriminant of the Weierstrass elliptic function (Apostol 1997, p. 130).

 

Hecke Algebra, Modular Form