viii Contents
§3.4. Hecke Operators 47
§3.5. Computing the Boundary Map 51
§3.6. Computing a Basis for S2(Γ0(N)) 53
§3.7. Computing S2(Γ0(N)) Using Eigenvectors 58
§3.8. Exercises 60
Chapter 4. Dirichlet Characters 63
§4.1. The Definition 64
§4.2. Representing Dirichlet Characters 64
§4.3. Evaluation of Dirichlet Characters 67
§4.4. Conductors of Dirichlet Characters 70
§4.5. The Kronecker Symbol 72
§4.6. Restriction, Extension, and Galois Orbits 75
§4.7. Alternative Representations of Characters 77
§4.8. Dirichlet Characters in SAGE 78
§4.9. Exercises 81
Chapter 5. Eisenstein Series and Bernoulli Numbers 83
§5.1. The Eisenstein Subspace 83
§5.2. Generalized Bernoulli Numbers 83
§5.3. Explicit Basis for the Eisenstein Subspace 88
§5.4. Exercises 90
Chapter 6. Dimension Formulas 91
§6.1. Modular Forms for Γ0(N) 92
§6.2. Modular Forms for Γ1(N) 95
§6.3. Modular Forms with Character 98
§6.4. Exercises 102
Chapter 7. Linear Algebra 103
§7.1. Echelon Forms of Matrices 103
§7.2. Rational Reconstruction 105
§7.3. Echelon Forms over Q 107
§7.4. Echelon Forms via Matrix Multiplication 110
§7.5. Decomposing Spaces under the Action of Matrix 114
§7.6. Exercises 119
Chapter 8. General Modular Symbols 121
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