According to Eichler there are five fundamental operations in arithmetic: "addition, subtraction, multiplication, division, and modular forms". This course is an introduction to modular forms at the first-year graduate level, with a special emphasis on how to compute with these objects.

Textbook. The main text is the book Computing with Modular Forms that I am finishing up, and which should be published by the AMS after the course is over. Four student projects from when I taught a similar course at Harvard and UCSD (http://modular.ucsd.edu:9000/classes/257) will be part of this book, and maybe your project will be too if you take this course.

• Modular forms of level 1
• Modular forms of higher level
• Eisenstein series, Bernoulli numbers, and Dirichlet characters
• Modular curves, elliptic curves, and Andrew Wiles's celebrated proof of Fermat's Last Theorem
• Computing modular forms using modular symbols
• L-functions and the the Birch and Swinnerton-Dyer conjecture

• Serre's conjectures (statement; computational questions; progress)
• Quaternion algebras and modular forms