12.3 Modular symbols and Hecke operators

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: 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

Here t2 represents the Hecke operator $ T_2$ on the space of Full Modular Symbols for $ \Gamma_0(1)$ of weight $ 12$ with sign 0 and dimension $ 3$ over $ \mathbb{Q}$ .

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,(3,2)], [X,(4,5)], [X,(5,4)])
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]

See the section on modular forms in the Sage Tutorial or the Reference Manual for more examples.

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