def string_to_list_of_solutions(s):
r"""
Used internally by the symbolic solve command to convert the output
of Maxima's solve command to a list of solutions in Sage's symbolic
package.
EXAMPLES:
We derive the (monic) quadratic formula::
sage: var('x,a,b')
(x, a, b)
sage: solve(x^2 + a*x + b == 0, x)
[x == -1/2*a - 1/2*sqrt(a^2 - 4*b), x == -1/2*a + 1/2*sqrt(a^2 - 4*b)]
Behind the scenes when the above is evaluated the function
:func:`string_to_list_of_solutions` is called with input the
string `s` below::
sage: s = '[x=-(sqrt(a^2-4*b)+a)/2,x=(sqrt(a^2-4*b)-a)/2]'
sage: sage.symbolic.relation.string_to_list_of_solutions(s)
[x == -1/2*a - 1/2*sqrt(a^2 - 4*b), x == -1/2*a + 1/2*sqrt(a^2 - 4*b)]
"""
from sage.categories.all import Objects
from sage.structure.sequence import Sequence
from sage.calculus.calculus import symbolic_expression_from_maxima_string
v = symbolic_expression_from_maxima_string(s, equals_sub=True)
return Sequence(v, universe=Objects(), cr_str=True)
def eisenstein_series(self):
"""
Return the Eisenstein series that span this space (over the
algebraic closure).
EXAMPLES::
sage: EisensteinForms(11,2).eisenstein_series()
[
5/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]
sage: EisensteinForms(1,4).eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6)
]
sage: EisensteinForms(1,24).eisenstein_series()
[
236364091/131040 + q + 8388609*q^2 + 94143178828*q^3 + 70368752566273*q^4 + 11920928955078126*q^5 + O(q^6)
]
sage: EisensteinForms(5,4).eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6),
1/240 + q^5 + O(q^6)
]
sage: EisensteinForms(13,2).eisenstein_series()
[
1/2 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]
sage: E = EisensteinForms(Gamma1(7),2)
sage: E.set_precision(4)
sage: E.eisenstein_series()
[
1/4 + q + 3*q^2 + 4*q^3 + O(q^4),
1/7*zeta6 - 3/7 + q + (-2*zeta6 + 1)*q^2 + (3*zeta6 - 2)*q^3 + O(q^4),
q + (-zeta6 + 2)*q^2 + (zeta6 + 2)*q^3 + O(q^4),
-1/7*zeta6 - 2/7 + q + (2*zeta6 - 1)*q^2 + (-3*zeta6 + 1)*q^3 + O(q^4),
q + (zeta6 + 1)*q^2 + (-zeta6 + 3)*q^3 + O(q^4)
]
sage: eps = DirichletGroup(13).0^2
sage: ModularForms(eps,2).eisenstein_series()
[
-7/13*zeta6 - 11/13 + q + (2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (6*zeta6 - 3)*q^4 - 4*q^5 + O(q^6),
q + (zeta6 + 2)*q^2 + (-zeta6 + 3)*q^3 + (3*zeta6 + 3)*q^4 + 4*q^5 + O(q^6)
]
sage: M = ModularForms(19,3).eisenstein_subspace()
sage: M.eisenstein_series()
[
]
sage: M = ModularForms(DirichletGroup(13).0, 1)
sage: M.eisenstein_series()
[
-1/13*zeta12^3 + 6/13*zeta12^2 + 4/13*zeta12 + 2/13 + q + (zeta12 + 1)*q^2 + zeta12^2*q^3 + (zeta12^2 + zeta12 + 1)*q^4 + (-zeta12^3 + 1)*q^5 + O(q^6)
]
sage: M = ModularForms(GammaH(15, [4]), 4)
sage: M.eisenstein_series()
[
1/240 + q + 9*q^2 + 28*q^3 + 73*q^4 + 126*q^5 + O(q^6),
1/240 + q^3 + O(q^6),
1/240 + q^5 + O(q^6),
1/240 + O(q^6),
1 + q - 7*q^2 - 26*q^3 + 57*q^4 + q^5 + O(q^6),
1 + q^3 + O(q^6),
q + 7*q^2 + 26*q^3 + 57*q^4 + 125*q^5 + O(q^6),
q^3 + O(q^6)
]
"""
P = self.parameters()
E = Sequence([
element.EisensteinSeries(self.change_ring(chi.base_ring()), None,
t, chi, psi) for chi, psi, t in P
],
immutable=True,
cr=True,
universe=Objects())
assert len(
E) == self.dimension(), "bug in enumeration of Eisenstein series."
return E