EXAMPLES:
sage: M = ModularForms(Gamma1(13),2); M
Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: S = M.cuspidal_submodule(); S
Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: S.basis()
[
q - 4*q^3 - q^4 + 3*q^5 + O(q^6),
q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6)
]
TESTS:
sage: m = ModularForms(Gamma1(20),2)
sage: loads(dumps(m)) == m
True
A space of modular forms for the group over the rational numbers.
Create a space of modular forms for of integral weight over the
rational numbers.
EXAMPLES:
sage: m = ModularForms(Gamma1(100),5); m
Modular Forms space of dimension 1270 for Congruence Subgroup Gamma1(100) of weight 5 over Rational Field
sage: type(m)
<class 'sage.modular.modform.ambient_g1.ModularFormsAmbient_g1_Q'>
Compute the matrix of the Hecke operator T_n acting on this space.
EXAMPLE:
sage: ModularForms(Gamma1(7), 4).hecke_matrix(3) # indirect doctest
[ 0 -42 133 0 0 0 0 0 0]
[ 0 -28 91 0 0 0 0 0 0]
[ 1 -8 19 0 0 0 0 0 0]
[ 0 0 0 28 0 0 0 0 0]
[ 0 0 0 -10152/259 0 5222/37 -13230/37 -22295/37 92504/37]
[ 0 0 0 -6087/259 0 312067/4329 1370420/4329 252805/333 3441466/4329]
[ 0 0 0 -729/259 1 485/37 3402/37 5733/37 7973/37]
[ 0 0 0 729/259 0 -189/37 -1404/37 -2366/37 -3348/37]
[ 0 0 0 255/259 0 -18280/4329 -51947/4329 -10192/333 -190855/4329]
Return the cuspidal submodule of this modular forms space.
EXAMPLES:
sage: m = ModularForms(Gamma1(17),2); m
Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(17) of weight 2 over Rational Field
sage: m.cuspidal_submodule()
Cuspidal subspace of dimension 5 of Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(17) of weight 2 over Rational Field
Return the Eisenstein submodule of this modular forms space.
EXAMPLES:
sage: ModularForms(Gamma1(13),2).eisenstein_submodule()
Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: ModularForms(Gamma1(13),10).eisenstein_submodule()
Eisenstein subspace of dimension 12 of Modular Forms space of dimension 69 for Congruence Subgroup Gamma1(13) of weight 10 over Rational Field