# Examples

See the [website] for many specific examples.

Ramanujan's tau-function defined implicitly by

also yields the simplest cusp form. The associated Fourier series satisfies

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)

and

where is the sum of the th powers of the positive divisors of :

Then, 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

where .

An example is the L-function associated to an elliptic curve 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.

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