def _multiple_of_constant(n, pos, const):
"""
Function for internal use in formatting ticks on axes with
nice-looking multiples of various symbolic constants, such
as `\pi` or `e`. Should only be used via keyword argument
`tick_formatter` in :meth:`plot.show`. See documentation
for the matplotlib.ticker module for more details.
EXAMPLES:
Here is the intended use::
sage: plot(sin(x), (x,0,2*pi), ticks=pi/3, tick_formatter=pi)
Graphics object consisting of 1 graphics primitive
Here is an unintended use, which yields unexpected (and probably
undesired) results::
sage: plot(x^2, (x, -2, 2), tick_formatter=pi)
Graphics object consisting of 1 graphics primitive
We can also use more unusual constant choices::
sage: plot(ln(x), (x,0,10), ticks=e, tick_formatter=e)
Graphics object consisting of 1 graphics primitive
sage: plot(x^2, (x,0,10), ticks=[sqrt(2),8], tick_formatter=sqrt(2))
Graphics object consisting of 1 graphics primitive
"""
from sage.misc.latex import latex
from sage.rings.arith import convergents
c = [i for i in convergents(n / const.n()) if i.denominator() < 12]
return '$%s$' % latex(c[-1] * const)
def _multiple_of_constant(n,pos,const):
"""
Function for internal use in formatting ticks on axes with
nice-looking multiples of various symbolic constants, such
as `\pi` or `e`. Should only be used via keyword argument
`tick_formatter` in :meth:`plot.show`. See documentation
for the matplotlib.ticker module for more details.
EXAMPLES:
Here is the intended use::
sage: plot(sin(x), (x,0,2*pi), ticks=pi/3, tick_formatter=pi)
Graphics object consisting of 1 graphics primitive
Here is an unintended use, which yields unexpected (and probably
undesired) results::
sage: plot(x^2, (x, -2, 2), tick_formatter=pi)
Graphics object consisting of 1 graphics primitive
We can also use more unusual constant choices::
sage: plot(ln(x), (x,0,10), ticks=e, tick_formatter=e)
Graphics object consisting of 1 graphics primitive
sage: plot(x^2, (x,0,10), ticks=[sqrt(2),8], tick_formatter=sqrt(2))
Graphics object consisting of 1 graphics primitive
"""
from sage.misc.latex import latex
from sage.rings.arith import convergents
c=[i for i in convergents(n/const.n()) if i.denominator()<12]
return '$%s$'%latex(c[-1]*const)
def unimod_matrices_from_infty(self,r,s):
"""
Returns a list of matrices whose associated unimodular paths connect r/s to infty.
(This is Manin's continued fraction trick.)
INPUT:
r,s -- rational numbers
OUTPUT:
a list of SL_2(Z) matrices
EXAMPLES:
"""
if s != 0:
v = []
# the function contfrac_q in https://github.com/williamstein/psage/blob/master/psage/modform/rational/modular_symbol_map.pyx
# is very, very relevant to massively optimizing this.
list = convergents(QQ(r)/s)
for j in range(0,len(list)-1):
a = list[j].numerator()
c = list[j].denominator()
b = list[j+1].numerator()
d = list[j+1].denominator()
v = v + [self.flip(Matrix(ZZ,[[(-1)**(j+1)*a,b],[(-1)**(j+1)*c,d]]))]
return [self.flip(Matrix(ZZ,[[1,list[0].numerator()],[0,list[0].denominator()]]))]+v
else:
return []
def unimod_matrices_to_infty(r, s):
r"""
Return
a list of matrices whose associated unimodular paths connect `0` to ``r/s``.
INPUT:
- ``r``, ``s`` -- rational numbers
OUTPUT:
- a list of matrices in `SL_2(\ZZ)`
EXAMPLES::
sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(19,23); v
[
[1 0] [ 0 1] [1 4] [-4 5] [ 5 19]
[0 1], [-1 1], [1 5], [-5 6], [ 6 23]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]
sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(11,25)
[
[1 0] [ 0 1] [1 3] [-3 4] [ 4 11]
[0 1], [-1 2], [2 7], [-7 9], [ 9 25]
]
ALGORITHM:
This is Manin's continued fraction trick, which gives an expression
`{0,r/s} = {0,\infty} + ... + {a,b} + ... + {*,r/s}`, where each `{a,b}` is
the image of `{0,\infty}` under a matrix in `SL_2(\ZZ)`.
"""
if s == 0:
return []
# the function contfrac_q in
# https://github.com/williamstein/psage/blob/master/psage/modform/rational/modular_symbol_map.pyx
# is very, very relevant to massively optimizing this.
L = convergents(r / s)
# Computes the continued fraction convergents of r/s
v = [M2Z([1, L[0].numerator(), 0, L[0].denominator()])]
# Initializes the list of matrices
for j in range(0, len(L)-1):
a = L[j].numerator()
c = L[j].denominator()
b = L[j + 1].numerator()
d = L[j + 1].denominator()
v.append(M2Z([(-1)**(j + 1) * a, b, (-1)**(j + 1) * c, d]))
# The matrix connecting two consecutive convergents is added on
return v
def unimod_matrices_to_infty(r, s):
r"""
Return a list of matrices whose associated unimodular paths connect `0` to ``r/s``.
INPUT:
- ``r``, ``s`` -- rational numbers
OUTPUT:
- a list of matrices in `SL_2(\ZZ)`
EXAMPLES::
sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(19,23); v
[
[1 0] [ 0 1] [1 4] [-4 5] [ 5 19]
[0 1], [-1 1], [1 5], [-5 6], [ 6 23]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]
sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_to_infty(11,25)
[
[1 0] [ 0 1] [1 3] [-3 4] [ 4 11]
[0 1], [-1 2], [2 7], [-7 9], [ 9 25]
]
ALGORITHM:
This is Manin's continued fraction trick, which gives an expression
`{0,r/s} = {0,\infty} + ... + {a,b} + ... + {*,r/s}`, where each `{a,b}` is
the image of `{0,\infty}` under a matrix in `SL_2(\ZZ)`.
"""
if s == 0:
return []
# the function contfrac_q in
# https://github.com/williamstein/psage/blob/master/psage/modform/rational/modular_symbol_map.pyx
# is very, very relevant to massively optimizing this.
L = convergents(r / s)
# Computes the continued fraction convergents of r/s
v = [M2Z([1, L[0].numerator(), 0, L[0].denominator()])]
# Initializes the list of matrices
for j in range(0, len(L)-1):
a = L[j].numerator()
c = L[j].denominator()
b = L[j + 1].numerator()
d = L[j + 1].denominator()
v.append(M2Z([(-1)**(j + 1) * a, b, (-1)**(j + 1) * c, d]))
# The matrix connecting two consecutive convergents is added on
return v
def unimod_matrices_from_infty(r, s):
r"""
Return a list of matrices whose associated unimodular paths connect `\infty` to ``r/s``.
INPUT:
- ``r``, ``s`` -- rational numbers
OUTPUT:
- a list of `SL_2(\ZZ)` matrices
EXAMPLES::
sage: v = sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(19,23); v
[
[ 0 1] [-1 0] [-4 1] [-5 -4] [-19 5]
[-1 0], [-1 -1], [-5 1], [-6 -5], [-23 6]
]
sage: [a.det() for a in v]
[1, 1, 1, 1, 1]
sage: sage.modular.pollack_stevens.manin_map.unimod_matrices_from_infty(11,25)
[
[ 0 1] [-1 0] [-3 1] [-4 -3] [-11 4]
[-1 0], [-2 -1], [-7 2], [-9 -7], [-25 9]
]
ALGORITHM:
This is Manin's continued fraction trick, which gives an expression
`{\infty,r/s} = {\infty,0} + ... + {a,b} + ... + {*,r/s}`, where each
`{a,b}` is the image of `{0,\infty}` under a matrix in `SL_2(\ZZ)`.
"""
if s != 0:
L = convergents(r / s)
# Computes the continued fraction convergents of r/s
v = [M2Z([-L[0].numerator(), 1, -L[0].denominator(), 0])]
# Initializes the list of matrices
# the function contfrac_q in https://github.com/williamstein/psage/blob/master/psage/modform/rational/modular_symbol_map.pyx
# is very, very relevant to massively optimizing this.
for j in range(0, len(L) - 1):
a = L[j].numerator()
c = L[j].denominator()
b = L[j + 1].numerator()
d = L[j + 1].denominator()
v.append(M2Z([-b, (-1)**(j + 1) * a, -d, (-1)**(j + 1) * c]))
# The matrix connecting two consecutive convergents is added on
return v
else:
return []
def __manin_symbol_rep(self, alpha):
"""
Return Manin symbol representation of X^i*Y^(k-2-i){0,alpha}.
EXAMPLES::
sage: s = ModularSymbols(11,2).1.modular_symbol_rep()[0][1]; s
{-1/8, 0}
sage: s.manin_symbol_rep() # indirect doctest
-(-8,1) - (1,1)
sage: M = ModularSymbols(11,2)
sage: s = M( (1,9) ); s
(1,9)
sage: t = s.modular_symbol_rep()[0][1].manin_symbol_rep(); t
-(-9,1) - (1,1)
sage: M(t)
(1,9)
"""
space = self.__space
i = self.__i
k = space.weight()
v = [(0, 1), (1, 0)]
if not alpha.is_infinity():
v += [(x.numerator(), x.denominator())
for x in arith.convergents(alpha._rational_())]
sign = 1
apply = sage.modular.modsym.manin_symbols.apply_to_monomial
mansym = sage.modular.modsym.manin_symbols.ManinSymbol
z = formal_sum.FormalSum(0)
for j in range(1, len(v)):
c = sign * v[j][1]
d = v[j - 1][1]
coeffs = apply(i, k - 2, sign * v[j][0], v[j - 1][0],
sign * v[j][1], v[j - 1][1])
w = [(coeffs[j], mansym(space, (j, c, d))) \
for j in range(k-1) if coeffs[j] != 0]
z += formal_sum.FormalSum(w)
sign *= -1
return z
def __manin_symbol_rep(self, alpha):
"""
Return Manin symbol representation of X^i*Y^(k-2-i){0,alpha}.
EXAMPLES::
sage: s = ModularSymbols(11,2).1.modular_symbol_rep()[0][1]; s
{-1/8, 0}
sage: s.manin_symbol_rep() # indirect doctest
-(-8,1) - (1,1)
sage: M = ModularSymbols(11,2)
sage: s = M( (1,9) ); s
(1,9)
sage: t = s.modular_symbol_rep()[0][1].manin_symbol_rep(); t
-(-9,1) - (1,1)
sage: M(t)
(1,9)
"""
space = self.__space
i = self.__i
k = space.weight()
v = [(0,1), (1,0)]
if not alpha.is_infinity():
v += [(x.numerator(), x.denominator()) for x in arith.convergents(alpha._rational_())]
sign = 1
apply = sage.modular.modsym.manin_symbols.apply_to_monomial
mansym = sage.modular.modsym.manin_symbols.ManinSymbol
z = formal_sum.FormalSum(0)
for j in range(1,len(v)):
c = sign*v[j][1]
d = v[j-1][1]
coeffs = apply(i, k-2, sign*v[j][0], v[j-1][0], sign*v[j][1], v[j-1][1])
w = [(coeffs[j], mansym(space, (j, c, d))) \
for j in range(k-1) if coeffs[j] != 0]
z += formal_sum.FormalSum(w)
sign *= -1
return z
