### Modular forms

Recall the formal product in the variable q we considered above. If we replace q with exp(2πiz), where z is a complex number, we obtain a function of z, which is called the Ramanujan Δ Function or discriminant function:_{n}are called the Fourier coefficients of f, and if the constant term a

_{0}is zero, then f is called a cusp form.

k | dim M_{k} | dim S_{k} |

2 | 0 | 0 |

4 | 1 | 0 |

6 | 1 | 0 |

8 | 1 | 0 |

10 | 1 | 0 |

12 | 2 | 1 |

14 | 1 | 0 |

16 | 2 | 1 |

18 | 2 | 1 |

20 | 2 | 1 |

22 | 2 | 1 |

24 | 3 | 2 |

_{k}of cusp forms of weight k is finite-dimensional. Of course, the same is then true for the subspace S

_{k}of cusp forms. If the Fourier coefficients of a cusp form f have the property that a

_{m}a

_{n}=a

_{mn}whenever gcd(m,n)=1, then f is called an eigenform. It can be proved that S

_{k}has a vector space basis consisting of eigenforms. The Sato-Tate conjecture is actually a statement for eigen-cuspforms, in the following sense. Let a

_{n}be the Fourier coefficients of a cusp form of weight k that is also an eigenform. By Deligne's theorem, we have

_{p}between 0 and π for which

More to come - go to page 3.