def rotation_list(tilt, turn):
from sage.functions.all import sin, cos
return [
sin(tilt * pi / 180.0),
sin(turn * pi / 180.0),
cos(tilt * pi / 180.0),
cos(turn * pi / 180.0)
]
def lam_minpoly(self):
r"""
Return the minimal polynomial of the corresponding lambda parameter of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(10).lam_minpoly()
x^4 - 5*x^2 + 5
sage: HeckeTriangleGroup(17).lam_minpoly()
x^8 - x^7 - 7*x^6 + 6*x^5 + 15*x^4 - 10*x^3 - 10*x^2 + 4*x + 1
sage: HeckeTriangleGroup(infinity).lam_minpoly()
x - 2
"""
# TODO: Write an explicit (faster) implementation
lam_symbolic = 2 * cos(pi / self._n)
return coerce_AA(lam_symbolic).minpoly()
def lam_minpoly(self):
r"""
Return the minimal polynomial of the corresponding lambda parameter of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: HeckeTriangleGroup(10).lam_minpoly()
x^4 - 5*x^2 + 5
sage: HeckeTriangleGroup(17).lam_minpoly()
x^8 - x^7 - 7*x^6 + 6*x^5 + 15*x^4 - 10*x^3 - 10*x^2 + 4*x + 1
sage: HeckeTriangleGroup(infinity).lam_minpoly()
x - 2
"""
# TODO: Write an explicit (faster) implementation
lam_symbolic = 2*cos(pi/self._n)
return coerce_AA(lam_symbolic).minpoly()
def __init__(self, n):
r"""
Hecke triangle group (2, n, infinity).
Namely the von Dyck group corresponding to the triangle group
with angles (pi/2, pi/n, 0).
INPUT:
- ``n`` - ``infinity`` or an integer greater or equal to ``3``.
OUTPUT:
The Hecke triangle group for the given parameter ``n``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(12)
sage: G
Hecke triangle group for n = 12
sage: G.category()
Category of groups
"""
self._n = n
self.element_repr_method("default")
if n in [3, infinity]:
self._base_ring = ZZ
self._lam = ZZ(1) if n == 3 else ZZ(2)
else:
lam_symbolic = 2 * cos(pi / n)
K = NumberField(self.lam_minpoly(),
'lam',
embedding=coerce_AA(lam_symbolic))
#self._base_ring = K.order(K.gens())
self._base_ring = K.maximal_order()
self._lam = self._base_ring.gen(1)
T = matrix(self._base_ring, [[1, self._lam], [0, 1]])
S = matrix(self._base_ring, [[0, -1], [1, 0]])
FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(2),
self._base_ring, [S, T])
def lam(self):
r"""
Return the parameter ``lambda`` of ``self``,
where ``lambda`` is twice the real part of ``rho``,
lying between ``1`` (when ``n=3``) and ``2`` (when ``n=infinity``).
EXAMPLES::
sage: HeckeTriangleGroup(3).lam()
1
sage: HeckeTriangleGroup(4).lam()^2
2.000000000000000?
sage: HeckeTriangleGroup(6).lam()^2
3.000000000000000?
sage: HeckeTriangleGroup(10).lam()
1.902113032590308?
"""
return AA(2 * cos(pi/self._n))
def __init__(self, n):
r"""
Hecke triangle group (2, n, infinity).
Namely the von Dyck group corresponding to the triangle group
with angles (pi/2, pi/n, 0).
INPUT:
- ``n`` - ``infinity`` or an integer greater or equal to ``3``.
OUTPUT:
The Hecke triangle group for the given parameter ``n``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.hecke_triangle_groups import HeckeTriangleGroup
sage: G = HeckeTriangleGroup(12)
sage: G
Hecke triangle group for n = 12
sage: G.category()
Category of groups
"""
self._n = n
self.element_repr_method("default")
if n in [3, infinity]:
self._base_ring = ZZ
self._lam = ZZ(1) if n==3 else ZZ(2)
else:
lam_symbolic = 2*cos(pi/n)
K = NumberField(self.lam_minpoly(), 'lam', embedding = coerce_AA(lam_symbolic))
#self._base_ring = K.order(K.gens())
self._base_ring = K.maximal_order()
self._lam = self._base_ring.gen(1)
T = matrix(self._base_ring, [[1,self._lam],[0,1]])
S = matrix(self._base_ring, [[0,-1],[1,0]])
FinitelyGeneratedMatrixGroup_generic.__init__(self, ZZ(2), self._base_ring, [S, T])
def dct(self):
"""
A discrete Cosine transform.
EXAMPLES::
sage: J = list(range(5))
sage: A = [exp(-2*pi*i*I/5) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dct()
Indexed sequence: [1/16*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + ...
indexed by [0, 1, 2, 3, 4]
"""
from sage.symbolic.constants import pi
F = self.base_ring() ## elements must be coercible into RR
J = self.index_object() ## must be = range(N)
N = len(J)
S = self.list()
PI = F(pi)
FT = [sum([S[i]*cos(2*PI*i/N) for i in J]) for j in J]
return IndexedSequence(FT,J)
def dct(self):
"""
Implements a discrete Cosine transform
EXAMPLES:
sage: J = range(5)
sage: A = [exp(-2*pi*i*I/5) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dct()
<BLANKLINE>
Indexed sequence: [1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1]
indexed by [0, 1, 2, 3, 4]
"""
from sage.symbolic.constants import pi
F = self.base_ring() ## elements must be coercible into RR
J = self.index_object() ## must be = range(N)
N = len(J)
S = self.list()
PI = F(pi)
FT = [sum([S[i]*cos(2*PI*i/N) for i in J]) for j in J]
return IndexedSequence(FT,J)
def dct(self):
"""
A discrete Cosine transform.
EXAMPLES::
sage: J = list(range(5))
sage: A = [exp(-2*pi*i*I/5) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dct()
Indexed sequence: [1/16*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + ...
indexed by [0, 1, 2, 3, 4]
"""
from sage.symbolic.constants import pi
F = self.base_ring() ## elements must be coercible into RR
J = self.index_object() ## must be = range(N)
N = len(J)
S = self.list()
PI = F(pi)
FT = [sum([S[i]*cos(2*PI*i/N) for i in J]) for j in J]
return IndexedSequence(FT,J)
def dct(self):
"""
Implements a discrete Cosine transform
EXAMPLES:
sage: J = range(5)
sage: A = [exp(-2*pi*i*I/5) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dct()
<BLANKLINE>
Indexed sequence: [1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1, 1/4*(sqrt(5) - 1)*e^(-8/5*I*pi) + 1/4*(sqrt(5) - 1)*e^(-2/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-6/5*I*pi) - 1/4*(sqrt(5) + 1)*e^(-4/5*I*pi) + 1]
indexed by [0, 1, 2, 3, 4]
"""
from sage.symbolic.constants import pi
F = self.base_ring() ## elements must be coercible into RR
J = self.index_object() ## must be = range(N)
N = len(J)
S = self.list()
PI = F(pi)
FT = [sum([S[i]*cos(2*PI*i/N) for i in J]) for j in J]
return IndexedSequence(FT,J)
def rotation_list(tilt,turn):
from sage.functions.all import sin, cos
return [ sin(tilt*pi/180.0), sin(turn*pi/180.0), cos(tilt*pi/180.0), cos(turn*pi/180.0) ]