2.11.4 Modular Forms

Sage can do some computations related to modular forms, including dimensions, computing spaces of modular symbols, Hecke operators, and decompositions.

There are several functions available for computing dimensions of spaces of modular forms. For example,

sage: dimension_cusp_forms(Gamma0(11),2)
1
sage: dimension_cusp_forms(Gamma0(1),12)
1
sage: dimension_cusp_forms(Gamma1(389),2)
6112

Next we illustrate computation of Hecke operators on a space of modular symbols of level $ 1$ and weight $ 12$ .

sage: M = ModularSymbols(1,12)
sage: M.basis()
([X^8*Y^2,(0,0)], [X^9*Y,(0,0)], [X^10,(0,0)])
sage: t2 = M.T(2)
sage: t2
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1) 
of weight 12 with sign 0 over Rational Field
sage: t2.matrix()
[ -24    0    0]
[   0  -24    0]
[4860    0 2049]
sage: f = t2.charpoly('x'); f
x^3 - 2001*x^2 - 97776*x - 1180224
sage: factor(f)
(x - 2049) * (x + 24)^2
sage: M.T(11).charpoly('x').factor()
(x - 285311670612) * (x - 534612)^2

We can also create spaces for $ \Gamma_0(N)$ and $ \Gamma_1(N)$ .

sage: ModularSymbols(11,2)
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
 0 over Rational Field
sage: ModularSymbols(Gamma1(11),2)
Modular Symbols space of dimension 11 for Gamma_1(11) of weight 2 with 
sign 0 and over Rational Field
Let's compute some characteristic polynomials and $ q$ -expansions.
sage: M = ModularSymbols(Gamma1(11),2)
sage: M.T(2).charpoly('x')
x^11 - 8*x^10 + 20*x^9 + 10*x^8 - 145*x^7 + 229*x^6 + 58*x^5 - 360*x^4 
     + 70*x^3 - 515*x^2 + 1804*x - 1452
sage: M.T(2).charpoly('x').factor()
(x - 3) * (x + 2)^2 * (x^4 - 7*x^3 + 19*x^2 - 23*x + 11) 
        * (x^4 - 2*x^3 + 4*x^2 + 2*x + 11)
sage: S = M.cuspidal_submodule()
sage: S.T(2).matrix()
[-2  0]
[ 0 -2]
sage: S.q_expansion_basis(10)
[
    q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 + O(q^10)
]

We can even compute spaces of modular symbols with character.

sage: G = DirichletGroup(13)
sage: e = G.0^2
sage: M = ModularSymbols(e,2); M
Modular Symbols space of dimension 4 and level 13, weight 2, character 
[zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2
sage: M.T(2).charpoly('x').factor()
(x - 2*zeta6 - 1) * (x - zeta6 - 2) * (x + zeta6 + 1)^2
sage: S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 2 of Modular Symbols space of 
dimension 4 and level 13, weight 2, character [zeta6], sign 0, over 
Cyclotomic Field of order 6 and degree 2
sage: S.T(2).charpoly('x').factor()
(x + zeta6 + 1)^2
sage: S.q_expansion_basis(10)
[
q + (-zeta6 - 1)*q^2 + (2*zeta6 - 2)*q^3 + zeta6*q^4 + (-2*zeta6 + 1)*q^5 
  + (-2*zeta6 + 4)*q^6 + (2*zeta6 - 1)*q^8 - zeta6*q^9 + O(q^10)
]

Here is another example of how Sage can compute the action of Hecke operators on a space of modular forms.

sage: T = ModularForms(Gamma0(11),2)
sage: T
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of 
weight 2 over Rational Field
sage: T.degree()
2
sage: T.level()
11
sage: T.group()
Congruence Subgroup Gamma0(11)
sage: T.dimension()
2
sage: T.cuspidal_subspace()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for
Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: T.eisenstein_subspace()
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2 
for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: M = ModularSymbols(11); M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
sage: M.weight()
2
sage: M.basis()
((1,0), (1,8), (1,9))
sage: M.sign()
0

Let $ T_p$ denote the usual Hecke operators ($ p$ prime). How do the Hecke operators $ T_2$ , $ T_3$ , $ T_5$ act on the space of modular symbols?

sage: M.T(2).matrix()
[ 3  0 -1]
[ 0 -2  0]
[ 0  0 -2]
sage: M.T(3).matrix()
[ 4  0 -1]
[ 0 -1  0]
[ 0  0 -1]
sage: M.T(5).matrix()
[ 6  0 -1]
[ 0  1  0]
[ 0  0  1]

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