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
on the
space of Full Modular Symbols for
of weight
with sign 0
and dimension
over
.
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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