Contents -- Acknowledgements -- Introduction -- This book -- Possible courses -- An overview of this book1 -- 1. Historical overview -- 1.1 18th Century - a prologue -- 1.2 19th century - the classical period -- 1.3 Early 20th century - arithmetic applications -- 1.4 Later 20th century - the link to elliptic curves -- 1.5 The 21st century - the Langlands Program -- 2. Introduction to modular forms -- 2.1 Modular forms for SL2(Z) -- 2.2 Eisenstein series for the full modular group -- 2.3 Computing Fourier expansions of Eisenstein series -- 2.4 Congruence subgroups -- 2.5 Fundamental domains -- 2.6 Modular forms for congruence subgroups -- 2.7 Eisenstein series for congruence subgroups -- 2.8 Derivatives of modular forms -- 2.8.1 Quasi-modular forms -- 2.9 Exercises -- 3. Results on finite-dimensionality -- 3.1 Spaces of modular forms are finite-dimensional -- 3.2 Explicit formulae for the dimensions of spaces of modular forms -- 3.2.1 Formulae for the full modular group -- 3.2.2 Formulae for congruence subgroups -- 3.3 The Sturm bound -- 3.4 Exercises -- 4. The arithmetic of modular forms -- 4.1 Hecke operators -- 4.1.1 Motivation for the Hecke operators -- 4.1.2 Hecke operators for Mk(SL2(Z)) -- 4.1.3 Hecke operators for congruence subgroups -- 4.2 Bases of eigenforms -- 4.2.1 The Petersson scalar product -- 4.2.2 The Hecke operators are Hermitian -- 4.2.3 Integral bases -- 4.3 Oldforms and newforms -- 4.3.1 Multiplicity one for newforms -- 4.4 Exercises -- 5. Applications of modular forms -- 5.1 Modular functions -- 5.2 η-products and η-quotients -- 5.3 The arithmetic of the j-invariant -- 5.3.1 The j-invariant and the Monster group -- 5.3.2 "Ramanujan's Constant" -- 5.4 Applications of the modular function λ(z) -- 5.4.1 Computing digits of π using λ(z) -- 5.4.2 Proving Picard's Theorem -- 5.5 Identities of series and products.

5.6 The Ramanujan-Petersson Conjecture -- 5.7 Elliptic curves and modular forms -- 5.7.1 Fermat's Last Theorem -- 5.8 Theta functions and their applications -- 5.8.1 Representations of n by a quadratic form in an even number of variables -- 5.8.2 Representations of n by a quadratic form in an odd number of variables -- 5.8.3 The Shimura correspondence -- 5.9 CM modular forms -- 5.10 Lacunary modular forms -- 5.11 Exercises -- 6. Modular forms in characteristic p -- 6.1 Classical treatment -- 6.1.1 The structure of the ring of mod p forms -- 6.1.2 The θ operator on mod p modular forms -- 6.1.3 Hecke operators and Hecke eigenforms -- 6.2 Galois representations attached to mod p modular forms -- 6.3 Katz modular forms -- 6.4 The Sturm bound in characteristic p -- 6.5 Computations with mod p modular forms -- 6.6 Exercises -- 7. Computing with modular forms -- 7.1 Historical introduction to computations in number theory -- 7.2 Magma -- 7.2.1 Magma philosophy -- 7.2.2 Magma programming -- 7.3 Sage -- 7.3.1 Sage philosophy -- 7.3.2 Sage programming -- 7.3.3 The Sage interface -- 7.3.4 Sage graphics -- 7.4 Pari and other systems -- 7.4.1 Pari -- 7.4.2 Other systems and solutions -- 7.5 Discussion of computation -- 7.5.1 Computation today -- 7.5.2 Expected running times -- 7.5.3 Using computation effectively -- 7.5.4 The limits of computation -- 7.5.4.1 Explicit examples of limitations -- 7.5.5 Guy's law of small numbers -- 7.5.6 How hard is it to calculate Fourier coe cients of modular forms? -- 7.6 Exercises -- 7.6.1 Magma -- 7.6.2 Sage -- 7.6.3 Pari -- 7.6.4 Maple -- Appendix A Magma code for classical modular forms -- Appendix B Sage code for classical modular forms -- Appendix C Hints and answers to selected exercises -- Bibliography -- List of Symbols -- Index.