Monstrous moonshine is the connection between the monster group 
and the j-function investigated by Conway and
Norton (1979). The Fourier series expansion of
is given by
 |
(1)
|
(OEIS A000521), where 
and 
is the half-period ratio
in the upper half-plane. The dimensions of the
first few irreducible representations
of 
are 1, 196883, 21296876, 842609326, ... (OEIS A001379).
J. McKay noticed in November 1978 that the 
-coefficient 196884 is exactly one more than the smallest
dimension of a nontrivial irreducible representation
of 
(Conway and Norton 1979). The first few Fourier series
coefficients of 
can be expressed as linear combinations of
these dimensions as follows:
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
Borcherds (1992) proved the monstrous moonshine conjectures of Conway and Norton (1979). Amazingly, there turn out to be yet more deep connections between the monster group and the j-function.
