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Author: William A. Stein
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SAGE Worksheet: general
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# Tutorial: Computing With Modular Forms Using Magma

Contents, General, Modular Forms, Modular Symbols, Future

# Congruence Subgroups

G := Gamma0(11); G
 Gamma_0(11) Gamma_0(11)
Generators(G)
 [ [1 1] [0 1], [ 3 -2] [11 -7], [ 4 -3] [11 -8] ] [ [1 1] [0 1], [ 3 -2] [11 -7], [ 4 -3] [11 -8] ]
CosetRepresentatives(G);
 [ [1 0] [0 1], [ 0 1] [-1 1], [-1 1] [-1 0], [1 0] [1 1], [ 0 1] [-1 2], [-1 1] [-2 1], [1 0] [2 1], [ 0 1] [-1 3], [-1 1] [-3 2], [1 1] [1 2], [-1 2] [-2 3], [-2 1] [-3 1] ] [ [1 0] [0 1], [ 0 1] [-1 1], [-1 1] [-1 0], [1 0] [1 1], [ 0 1] [-1 2], [-1 1] [-2 1], [1 0] [2 1], [ 0 1] [-1 3], [-1 1] [-3 2], [1 1] [1 2], [-1 2] [-2 3], [-2 1] [-3 1] ]

# Dimension Formulas

DimensionCuspFormsGamma0(11,2)
 1 1
DimensionCuspFormsGamma1(13,2)
 2 2
DimensionNewCuspFormsGamma0(100,2)
 1 1

# Dirichlet Characters

G<a> := DirichletGroup(37);
G
 Group of Dirichlet characters of modulus 37 over Rational Field Group of Dirichlet characters of modulus 37 over Rational Field
Order(a)
 2 2
[a(n) : n in [1..10]]
 [ 1, -1, 1, 1, -1, -1, 1, -1, 1, 1 ] [ 1, -1, 1, 1, -1, -1, 1, -1, 1, 1 ]
Eltseq(a)
 [ 18 ] [ 18 ]
DimensionCuspForms(a,2)
 2 2
G<a,b> := DirichletGroup(4*37, CyclotomicField(36));
G
 Group of Dirichlet characters of modulus 148 over Cyclotomic Field of order 36 and degree 12 Group of Dirichlet characters of modulus 148 over Cyclotomic Field of order 36 and degree 12
Eltseq(b)
 [ 0, 1 ] [ 0, 1 ]