- Title Pages
- Let τ:ℕ→ℤ be defined by
- Contents
- Preface
- Acknowledgments
- Author information
- Dependencies between the chapters
- Chapter One Introduction, main results, context
- Chapter Two Modular curves, modular forms, lattices, Galois representations
- Chapter Three First description of the algorithms
- Chapter Four Short introduction to heights and Arakelov theory
- Chapter Five Computing complex zeros of polynomials and power series
- Chapter Six Computations with modular forms and Galois representations
- Chapter Seven Polynomials for projective representations of level one forms
- Chapter Eight Description of <i>X</i><sub>1</sub>(5<i>l</i>)
- Chapter Nine Applying Arakelov theory
- Chapter Ten An upper bound for Green functions on Riemann surfaces
- Chapter Eleven Bounds for Arakelov invariants of modular curves
- Chapter Twelve Approximating <i>V<sub>f</sub></i> over the complex numbers
- Chapter Thirteen Computing <i>V<sub>f</sub></i> modulo <i>p</i>
- Chapter Fourteen Computing the residual Galois representations
- Chapter Fifteen Computing coefficients of modular forms
- Epilogue
- Bibliography
- Index

# Computing coefficients of modular forms

# Computing coefficients of modular forms

- Chapter:
- (p.383) Chapter Fifteen Computing coefficients of modular forms
- Source:
- Computational Aspects of Modular Forms and Galois Representations
- Author(s):
### Bas Edixhoven

- Publisher:
- Princeton University Press

This chapter applies the main result on the computation of Galois representations attached to modular forms of level one to the computation of coefficients of modular forms. It treats the case of the discriminant modular form, that is, the computation of Ramanujan's tau-function at primes, and then deals with the more general case of forms of level one and arbitrary weight *k*, reformulated as the computation of Hecke operators *T*ⁿ as ℤ-linear combinations of the *T*ᵢ with *i* < *k* = 12. The chapter gives an application to theta functions of even, unimodular positive definite quadratic forms over ℤ.

*Keywords:*
modular forms, Galois representations, coefficients, Ramanujan's tau-function, Hecke operators

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- Title Pages
- Let τ:ℕ→ℤ be defined by
- Contents
- Preface
- Acknowledgments
- Author information
- Dependencies between the chapters
- Chapter One Introduction, main results, context
- Chapter Two Modular curves, modular forms, lattices, Galois representations
- Chapter Three First description of the algorithms
- Chapter Four Short introduction to heights and Arakelov theory
- Chapter Five Computing complex zeros of polynomials and power series
- Chapter Six Computations with modular forms and Galois representations
- Chapter Seven Polynomials for projective representations of level one forms
- Chapter Eight Description of <i>X</i><sub>1</sub>(5<i>l</i>)
- Chapter Nine Applying Arakelov theory
- Chapter Ten An upper bound for Green functions on Riemann surfaces
- Chapter Eleven Bounds for Arakelov invariants of modular curves
- Chapter Twelve Approximating <i>V<sub>f</sub></i> over the complex numbers
- Chapter Thirteen Computing <i>V<sub>f</sub></i> modulo <i>p</i>
- Chapter Fourteen Computing the residual Galois representations
- Chapter Fifteen Computing coefficients of modular forms
- Epilogue
- Bibliography
- Index