 Title Pages
 Let $\tau :\mathbb{N}\to \mathbb{Z}$ 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 X_{1}(5l) 
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 V_{f} over the complex numbers 
Chapter Thirteen Computing V_{f} modulo p 
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 taufunction 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 Tⁿ as ℤlinear combinations of the Tᵢ 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 taufunction, Hecke operators
Princeton Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.
Please, subscribe or login to access full text content.
If you think you should have access to this title, please contact your librarian.
To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us.
 Title Pages
 Let $\tau :\mathbb{N}\to \mathbb{Z}$ 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 X_{1}(5l) 
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 V_{f} over the complex numbers 
Chapter Thirteen Computing V_{f} modulo p 
Chapter Fourteen Computing the residual Galois representations 
Chapter Fifteen Computing coefficients of modular forms  Epilogue
 Bibliography
 Index