def _second_gen_unramified(self):
r"""
Calculate the action of the matrix [0, -1; 1, 0] on the type space,
in the unramified (even level) case.
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: T = example_type_space(2)
sage: T._second_gen_unramified()
[ 0 1 -2 1]
[ 0 0 -1 0]
[ 0 -1 0 0]
[ 1 -2 1 0]
sage: T._second_gen_unramified()**4 == 1
True
"""
f = self.prime()**self.u()
g2 = lift_gen_to_gamma1(f, self.tame_level())
g3 = [f * g2[0], g2[1], f**2 * g2[2], f * g2[3]]
A = self.t_space.ambient()
mm = A._action_on_modular_symbols(g3).restrict(
self.t_space.free_module()).transpose()
m = mm / ZZ(f**(self.form().weight() - 2))
return m
def _second_gen_unramified(self):
r"""
Calculate the action of the matrix [0, -1; 1, 0] on the type space,
in the unramified (even level) case.
EXAMPLE::
sage: from sage.modular.local_comp.type_space import example_type_space
sage: T = example_type_space(2)
sage: T._second_gen_unramified()
[ 0 1 -2 1]
[ 0 0 -1 0]
[ 0 -1 0 0]
[ 1 -2 1 0]
sage: T._second_gen_unramified()**4 == 1
True
"""
f = self.prime() ** self.u()
g2 = lift_gen_to_gamma1(f, self.tame_level())
g3 = [f * g2[0], g2[1], f**2 * g2[2], f*g2[3]]
A = self.t_space.ambient()
mm = A._action_on_modular_symbols(g3).restrict(self.t_space.free_module()).transpose()
m = mm / ZZ(f**(self.form().weight()-2))
return m