See the [website]
for many specific examples.
Ramanujan's tau-function defined implicitly by
also yields the simplest cusp form. The associated Fourier series
for all integers with which means that it is a cusp form of weight 12 for the full modular group.
The unique cusp forms of weights 16, 18, 20, 22, and 26 for the full modular group can be given
explicitly in terms of (the Eisenstein series)
where is the sum of the th powers of the positive
divisors of :
gives the unique Hecke form of weight 16;
gives the unique Hecke form of weight 18;
is the Hecke form of weight 20;
is the Hecke form of weight 22; and
is the Hecke form of weight 26. The two
Hecke forms of weight 24 are given by
An example is the L-function associated to an elliptic
where are integers.
The associated L-function, called the Hasse-Weil L-function, is
where is the conductor of the curve. The coefficients
are constructed easily from for prime ; in turn the
are given by where is the number of
solutions of when considered modulo . The work of Wiles and
others proved that these L-functions are associated to modular
forms of weight 2.
Back to the
for The Riemann Hypothesis.