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

(1)

For  a prime , the operator collapses to

(2)

If  has the Fourier series

(3)

then  has Fourier series

(4)

where

(5)

(Apostol 1997, p. 121).

If , the Hecke operators obey the composition property

(6)

Any two Hecke operators  and  on  commute with each other, and moreover

(7)

(Apostol 1997, pp. 126-127).

Each Hecke operator  has eigenforms when the dimension of  is 1, so for , 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 , 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).

REFERENCES:

Apostol, T. M. "The Hecke Operators." §6.7 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 120-122, 1997.