Modular Forms

This section is about computing with modular forms, modular symbols, and modular abelian varieties. Most of the Sage functionality we describe below is new code written for Sage by myself, Craig Citro, Robert Bradshaw, and Jordi Quer in consultation with John Cremona. It has much overlap in functionality with the modular forms code in Magma, which I developed during 1998-2004.

• Modular Forms and Hecke Operators
  • Congruence subgroups
  • Modular Forms
• Modular Symbols
  • Definition
  • Manin Symbols
• Method of Graphs
• Level One Modular Forms
  • Computing \(\Delta\)
  • The Victor Miller Basis
• Half Integral Weight Forms
  • Basmaji’s Algorithm
• Generators for Rings of Modular Forms
  • Computing Generators
• \(L\)-series
  • \(L\)-series of \(\Delta\)
  • \(L\)-series of a Cusp Form
  • \(L\)-series of a General Newform is Not Implemented
• Modular Abelian Varieties
  • Computing in Sage
  • Endomorphisms