xiv Preface

symbols, Manin symbols, cohomology of subgroups of SL2(Z) with various

coeﬃcients, explicit computation of modular forms, etc.

Software. We use SAGE, Software for Algebra and Geometry Experimen-

tation (see [Ste06]), to illustrate how to do many of the examples. SAGE

is completely free and packages together a wide range of open source math-

ematics software for doing much more than just computing with modular

forms. SAGE can be downloaded and run on your computer or can be used

via a web browser over the Internet. The reader is encouraged to experi-

ment with many of the objects in this book using SAGE. We do not describe

the basics of using SAGE in this book; the reader should read the SAGE

tutorial (and other documentation) available at the SAGE website [Ste06].

All examples in this book have been automatically tested and should work

exactly as indicated in SAGE version at least 1.5.

Acknowledgements. David Joyner and Gabor Wiese carefully read the

book and provided a huge number of helpful comments.

John Cremona and Kevin Buzzard both made many helpful remarks that

were important in the development of the algorithms in this book. Much of

the mathematics (and some of the writing) in Chapter 10 is joint work with

Helena Verrill.

Noam Elkies made remarks about Chapters 1 and 2. S´ andor Kov´acs

provided interesting comments on Chapter 1. Allan Steel provided helpful

feedback on Chapter 7. Jordi Quer made useful remarks about Chapter 4

and Chapter 6.

The students in the courses that I taught on this material at Harvard,

San Diego, and Washington provided substantial feedback: in particular,

Abhinav Kumar made numerous observations about computing widths of

cusps (see Section 1.4.1) and Thomas James Barnet-Lamb made helpful re-

marks about how to represent Dirichlet characters. James Merryfield made

helpful remarks about complex analytic issues and about convergence in Stir-

ling’s formula. Robert Bradshaw, Andrew Crites (who wrote Exercise 7.5),

Michael Goff, Dustin Moody, and Koopa Koo wrote most of the solutions

included in Chapter 11 and found numerous typos throughout the book.

Dustin Moody also carefully read through the book and provided feedback.

H. Stark suggested using Stirling’s formula in Section 2.7.1, and Mark

Watkins and Lynn Walling made comments on Chapter 3.

Parts of Chapter 1 follow Serre’s beautiful introduction to modular forms

[Ser73, Ch. VII] closely, though we adjust the notation, definitions, and

order of presentation to be consistent with the rest of this book.