Contents Preface xi Chapter 1. Modular Forms 1 §1.1. Basic Definitions 1 §1.2. Modular Forms of Level 1 3 §1.3. Modular Forms of Any Level 4 §1.4. Remarks on Congruence Subgroups 7 §1.5. Applications of Modular Forms 9 §1.6. Exercises 11 Chapter 2. Modular Forms of Level 1 13 §2.1. Examples of Modular Forms of Level 1 13 §2.2. Structure Theorem for Level 1 Modular Forms 17 §2.3. The Miller Basis 20 §2.4. Hecke Operators 22 §2.5. Computing Hecke Operators 26 §2.6. Fast Computation of Fourier Coefficients 29 §2.7. Fast Computation of Bernoulli Numbers 29 §2.8. Exercises 33 Chapter 3. Modular Forms of Weight 2 35 §3.1. Hecke Operators 36 §3.2. Modular Symbols 39 §3.3. Computing with Modular Symbols 41 vii
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