模块化形式¶

尖点形状¶

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


陪侍代表¶

sage: G = Gamma0(11); G
Congruence Subgroup Gamma0(11)
sage: list(G.coset_reps())
[
[1 0]  [ 0 -1]  [1 0]  [ 0 -1]  [ 0 -1]  [ 0 -1]  [ 0 -1]  [ 0 -1]
[0 1], [ 1  0], [1 1], [ 1  2], [ 1  3], [ 1  4], [ 1  5], [ 1  6],
<BLANKLINE>
[ 0 -1]  [ 0 -1]  [ 0 -1]  [ 0 -1]
[ 1  7], [ 1  8], [ 1  9], [ 1 10]
]


模符号与Hecke算子¶

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: 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


sage: M = ModularSymbols(Gamma1(6),3,sign=0)
sage: M
Modular Symbols space of dimension 4 for Gamma_1(6) of weight 3 with sign 0
and over Rational Field
sage: M.basis()
([X,(0,5)], [X,(3,5)], [X,(4,5)], [X,(5,5)])
sage: M._compute_hecke_matrix_prime(2).charpoly()
x^4 - 17*x^2 + 16
sage: M.integral_structure()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]


亏格公式¶

Sage能计算出 $$X_0(N)$$$$X_1(N)$$ ，以及相关曲线。以下是一些语法示例：

sage: dimension_cusp_forms(Gamma0(22))
2
sage: dimension_cusp_forms(Gamma0(30))
3
sage: dimension_cusp_forms(Gamma1(30))
9


See the code for computing dimensions of spaces of modular forms (in sage/modular/dims.py) or the paper by Oesterlé and Cohen {CO} for some details.