Sharedwww / talks / 20070604-banff / general.htmlOpen in CoCalc
Author: William A. Stein
SAGE Worksheet: general
jsMath

Tutorial: Computing With Modular Forms Using Magma

Contents, General, Modular Forms, Modular Symbols, Future

General Modular Forms-Related Functionality



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 ]