| Acknowledgements | p. vii |
| Introduction | p. 1 |
| Historical overview | p. 5 |
| 18th Century - a prologue | p. 5 |
| 19th century - the classical period | p. 6 |
| Early 20th century - arithmetic applications | p. 7 |
| Later 20th century - the link to elliptic curves | p. 8 |
| The 21st century - the Langlands Program | p. 9 |
| Introduction to modular forms | p. 11 |
| Modular forms for SL[subscript 2](Z) | p. 11 |
| Eisenstein series for the full modular group | p. 15 |
| Computing Fourier expansions of Eisenstein series | p. 17 |
| Congruence subgroups | p. 21 |
| Fundamental domains | p. 25 |
| Modular forms for congruence subgroups | p. 28 |
| Eisenstein series for congruence subgroups | p. 32 |
| Derivatives of modular forms | p. 35 |
| Quasi-modular forms | p. 37 |
| Exercises | p. 38 |
| Results on finite-dimensionality | p. 41 |
| Spaces of modular forms are finite-dimensional | p. 41 |
| Explicit formulae for the dimensions of spaces of modular forms | p. 46 |
| Formulae for the full modular group | p. 46 |
| Formulae for congruence subgroups | p. 49 |
| The Sturm bound | p. 52 |
| Exercises | p. 55 |
| The arithmetic of modular forms | p. 57 |
| Hecke operators | p. 58 |
| Motivation for the Hecke operators | p. 58 |
| Hecke operators for M[subscript k](SL[subscript 2](Z)) | p. 59 |
| Hecke operators for congruence subgroups | p. 63 |
| Bases of eigenforms | p. 69 |
| The Petersson scalar product | p. 69 |
| The Hecke operators are Hermitian | p. 75 |
| Integral bases | p. 79 |
| Oldforms and newforms | p. 80 |
| Multiplicity one for newforms | p. 85 |
| Exercises | p. 88 |
| Applications of modular forms | p. 93 |
| Modular functions | p. 94 |
| [eta]-products and [eta]-quotients | p. 98 |
| The arithmetic of the j-invariant | p. 103 |
| The j-invariant and the Monster group | p. 106 |
| "Ramanujan's Constant" | p. 107 |
| Applications of the modular function [lambda](z) | p. 108 |
| Computing digits of [pi] using [lambda](z) | p. 109 |
| Proving Picard's Theorem | p. 111 |
| Identities of series and products | p. 112 |
| The Ramanujan-Petersson Conjecture | p. 113 |
| Elliptic curves and modular forms | p. 116 |
| Fermat's Last Theorem | p. 119 |
| Theta functions and their applications | p. 120 |
| Representations of n by a quadratic form in an even number of variables | p. 121 |
| Representations of n by a quadratic form in an odd number of variables | p. 128 |
| The Shimura correspondence | p. 131 |
| CM modular forms | p. 133 |
| Lacunary modular forms | p. 135 |
| Exercises | p. 138 |
| Modular forms in characteristic p | p. 143 |
| Classical treatment | p. 143 |
| The structure of the ring of mod p forms | p. 144 |
| The [theta] operator on mod p modular forms | p. 150 |
| Hecke operators and Hecke eigenforms | p. 151 |
| Galois representations attached to mod p modular forms | p. 152 |
| Katz modular forms | p. 156 |
| The Sturm bound in characteristic p | p. 158 |
| Computations with mod p modular forms | p. 159 |
| Exercises | p. 161 |
| Computing with modular forms | p. 163 |
| Historical introduction to computations in number theory | p. 163 |
| Magma | p. 167 |
| Magma philosophy | p. 170 |
| Magma programming | p. 171 |
| Sage | p. 173 |
| Sage philosophy | p. 175 |
| Sage programming | p. 175 |
| The Sage interface | p. 176 |
| Sage graphics | p. 177 |
| Pari and other systems | p. 177 |
| Pari | p. 177 |
| Other systems and solutions | p. 179 |
| Discussion of computation | p. 180 |
| Computation today | p. 180 |
| Expected running times | p. 182 |
| Using computation effectively | p. 183 |
| The limits of computation | p. 184 |
| Guy's law of small numbers | p. 187 |
| How hard is it to calculate Fourier coefficients of modular forms? | p. 189 |
| Exercises | p. 189 |
| Magma | p. 190 |
| Sage | p. 191 |
| Pari | p. 193 |
| Maple | p. 193 |
| Magma code for classical modular forms | p. 195 |
| Sage code for classical modular forms | p. 197 |
| Hints and answers to selected exercises | p. 199 |
| Bibliography | p. 205 |
| List of Symbols | p. 217 |
| Index | p. 221 |
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