## Modular Forms: A Classical and Computational IntroductionThis book presents a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to such diverse subjects as the theory of quadratic forms, the proof of Fermat's last theorem and the approximation of pi. It provides a balanced overview of both the theoretical and computational sides of the subject, allowing a variety of courses to be taught from it. |

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### Contents

5 | |

11 | |

3 Results on finitedimensionality | 41 |

4 The arithmetic of modular forms | 57 |

5 Applications of modular forms | 93 |

6 Modular forms in characteristic p | 143 |

7 Computing with modular forms | 163 |

### Common terms and phrases

algorithm anqn arithmetic basis of eigenforms Chapter compute congruence subgroup conjecture consider constant term cusp form cuspidal deﬁne deﬁnition diﬀerent dimension formulae Dirichlet character double coset E4 and E6 eigenforms Eisenstein series element elliptic curves example Exercise Fermat’s Last Theorem ﬁeld ﬁnd a basis ﬁnding ﬁnite ﬁnite-dimensional ﬁrst form f form of weight forms for SL2(Z Fourier coeﬃcients Fourier expansion function of weight fundamental domain given Gk(z Hecke algebra Hecke operators Hecke operators Tn inﬁnitely integer and let ISBN j-invariant Koblitz Let f Magma Math Mathematics matrix mod p modular modular forms modular function modular group modulo multiplication newform nonzero number theory oldforms positive integer prime number proof Proposition prove Ramanujan Ramanujan-Petersson conjecture representation result Sage Section space of modular Sturm bound theory of modular theta functions transforms correctly write zero