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).
