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).
