Heilbronn matrix computation#

class sage.modular.modsym.heilbronn.Heilbronn#

Bases: object

apply(u, v, N)[source]#

Return a list of pairs \(((c,d),m)\), which is obtained as follows:

1) Compute the images \((a,b)\) of the vector \((u,v) \pmod N\) acted on by each of the HeilbronnCremona matrices in self.

  1. Reduce each \((a,b)\) to canonical form \((c,d)\) using p1normalize.

  2. Sort.

4) Create the list \(((c,d),m)\), where \(m\) is the number of times that \((c,d)\) appears in the list created in steps 1-3 above. Note that the pairs \(((c,d),m)\) are sorted lexicographically by \((c,d)\).

INPUT:

  • u, v, N – integers

OUTPUT: list

EXAMPLES:

sage: H = sage.modular.modsym.heilbronn.HeilbronnCremona(2); H
The Cremona-Heilbronn matrices of determinant 2
sage: H.apply(1,2,7)
[((1, 1), 1), ((1, 4), 1), ((1, 5), 1), ((1, 6), 1)]

>>> from sage.all import *
>>> H = sage.modular.modsym.heilbronn.HeilbronnCremona(Integer(2)); H
The Cremona-Heilbronn matrices of determinant 2
>>> H.apply(Integer(1),Integer(2),Integer(7))
[((1, 1), 1), ((1, 4), 1), ((1, 5), 1), ((1, 6), 1)]
to_list()[source]#

Return the list of Heilbronn matrices corresponding to self.

Each matrix is given as a list of four integers.

EXAMPLES:

sage: H = HeilbronnCremona(2); H
The Cremona-Heilbronn matrices of determinant 2
sage: H.to_list()
[[1, 0, 0, 2], [2, 0, 0, 1], [2, 1, 0, 1], [1, 0, 1, 2]]

>>> from sage.all import *
>>> H = HeilbronnCremona(Integer(2)); H
The Cremona-Heilbronn matrices of determinant 2
>>> H.to_list()
[[1, 0, 0, 2], [2, 0, 0, 1], [2, 1, 0, 1], [1, 0, 1, 2]]
class sage.modular.modsym.heilbronn.HeilbronnCremona[source]#

Bases: Heilbronn

Create the list of Heilbronn-Cremona matrices of determinant \(p\).

EXAMPLES:

sage: H = HeilbronnCremona(3) ; H
The Cremona-Heilbronn matrices of determinant 3
sage: H.to_list()
[[1, 0, 0, 3],
[3, 1, 0, 1],
[1, 0, 1, 3],
[3, 0, 0, 1],
[3, -1, 0, 1],
[-1, 0, 1, -3]]

>>> from sage.all import *
>>> H = HeilbronnCremona(Integer(3)) ; H
The Cremona-Heilbronn matrices of determinant 3
>>> H.to_list()
[[1, 0, 0, 3],
[3, 1, 0, 1],
[1, 0, 1, 3],
[3, 0, 0, 1],
[3, -1, 0, 1],
[-1, 0, 1, -3]]
p[source]#
class sage.modular.modsym.heilbronn.HeilbronnMerel[source]#

Bases: Heilbronn

Initialize the list of Merel-Heilbronn matrices of determinant \(n\).

EXAMPLES:

sage: H = HeilbronnMerel(3) ; H
The Merel-Heilbronn matrices of determinant 3
sage: H.to_list()
[[1, 0, 0, 3],
[1, 0, 1, 3],
[1, 0, 2, 3],
[2, 1, 1, 2],
[3, 0, 0, 1],
[3, 1, 0, 1],
[3, 2, 0, 1]]

>>> from sage.all import *
>>> H = HeilbronnMerel(Integer(3)) ; H
The Merel-Heilbronn matrices of determinant 3
>>> H.to_list()
[[1, 0, 0, 3],
[1, 0, 1, 3],
[1, 0, 2, 3],
[2, 1, 1, 2],
[3, 0, 0, 1],
[3, 1, 0, 1],
[3, 2, 0, 1]]
n[source]#
sage.modular.modsym.heilbronn.hecke_images_gamma0_weight2(u, v, N, indices, R)[source]#

INPUT:

  • u, v, N – integers so that \(\gcd(u,v,N) = 1\)

  • indices – a list of positive integers

  • R – matrix over \(\QQ\) that writes each elements of \(\textnormal{P1} = \textnormal{P1List}(N)\) in terms of a subset of \(\textnormal{P1}\).

OUTPUT: a dense matrix whose columns are the images \(T_n(x)\) for \(n\) in indices and \(x\) the Manin symbol \((u,v)\), expressed in terms of the basis.

EXAMPLES:

sage: M = ModularSymbols(23,2,1)
sage: A = sage.modular.modsym.heilbronn.hecke_images_gamma0_weight2(1,0,23,[1..6],M.manin_gens_to_basis())
sage: rowsA = A.rows()
sage: z = M((1,0))
sage: all(M.T(n)(z).element() == rowsA[n-1] for n in [1..6])
True

>>> from sage.all import *
>>> M = ModularSymbols(Integer(23),Integer(2),Integer(1))
>>> A = sage.modular.modsym.heilbronn.hecke_images_gamma0_weight2(Integer(1),Integer(0),Integer(23),(ellipsis_range(Integer(1),Ellipsis,Integer(6))),M.manin_gens_to_basis())
>>> rowsA = A.rows()
>>> z = M((Integer(1),Integer(0)))
>>> all(M.T(n)(z).element() == rowsA[n-Integer(1)] for n in (ellipsis_range(Integer(1),Ellipsis,Integer(6))))
True
sage.modular.modsym.heilbronn.hecke_images_gamma0_weight_k(u, v, i, N, k, indices, R)[source]#

INPUT:

  • u, v, N – integers so that \(\gcd(u,v,N) = 1\)

  • i – integer with \(0 \le i \le k-2\)

  • k – weight

  • indices – a list of positive integers

  • R – matrix over \(\QQ\) that writes each elements of \(\textnormal{P1} = \textnormal{P1List}(N)\) in terms of a subset of \(\textnormal{P1}\).

OUTPUT: a dense matrix with rational entries whose columns are the images \(T_n(x)\) for \(n\) in indices and \(x\) the Manin symbol [\(X^i*Y^{(k-2-i)}\), \((u,v)\)], expressed in terms of the basis.

EXAMPLES:

sage: M = ModularSymbols(15,6,sign=-1)
sage: R = M.manin_gens_to_basis()
sage: a,b,c = sage.modular.modsym.heilbronn.hecke_images_gamma0_weight_k(4,1,3,15,6,[1,11,12], R)
sage: x = M((3,4,1)) ; x.element() == a
True
sage: M.T(11)(x).element() == b
True
sage: M.T(12)(x).element() == c
True

>>> from sage.all import *
>>> M = ModularSymbols(Integer(15),Integer(6),sign=-Integer(1))
>>> R = M.manin_gens_to_basis()
>>> a,b,c = sage.modular.modsym.heilbronn.hecke_images_gamma0_weight_k(Integer(4),Integer(1),Integer(3),Integer(15),Integer(6),[Integer(1),Integer(11),Integer(12)], R)
>>> x = M((Integer(3),Integer(4),Integer(1))) ; x.element() == a
True
>>> M.T(Integer(11))(x).element() == b
True
>>> M.T(Integer(12))(x).element() == c
True
sage.modular.modsym.heilbronn.hecke_images_nonquad_character_weight2(u, v, N, indices, chi, R)[source]#

Return images of the Hecke operators \(T_n\) for \(n\) in the list indices, where \(\chi\) must be a quadratic Dirichlet character with values in \(\QQ\).

\(R\) is assumed to be the relation matrix of a weight modular symbols space over \(\QQ\) with character \(\chi\).

INPUT:

  • u, v, N – integers so that \(\gcd(u,v,N) = 1\)

  • indices – a list of positive integers

  • chi – a Dirichlet character that takes values in a nontrivial extension of \(\QQ\).

  • R – matrix over \(\QQ\) that writes each elements of \(\textnormal{P1} = \textnormal{P1List}(N)\) in terms of a subset of \(\textnormal{P1}\).

OUTPUT: a dense matrix with entries in the field \(\QQ(\chi)\) (the values of \(\chi\)) whose columns are the images \(T_n(x)\) for \(n\) in indices and \(x\) the Manin symbol \((u,v)\), expressed in terms of the basis.

EXAMPLES:

sage: chi = DirichletGroup(13).0^2
sage: M = ModularSymbols(chi)
sage: eps = M.character()
sage: R = M.manin_gens_to_basis()
sage: sage.modular.modsym.heilbronn.hecke_images_nonquad_character_weight2(1,0,13,[1,2,6],eps,R)
[           1            0            0            0]
[   zeta6 + 2            0            0           -1]
[           7 -2*zeta6 + 1   -zeta6 - 1     -2*zeta6]
sage: x = M((1,0)); x.element()
(1, 0, 0, 0)
sage: M.T(2)(x).element()
(zeta6 + 2, 0, 0, -1)
sage: M.T(6)(x).element()
(7, -2*zeta6 + 1, -zeta6 - 1, -2*zeta6)

>>> from sage.all import *
>>> chi = DirichletGroup(Integer(13)).gen(0)**Integer(2)
>>> M = ModularSymbols(chi)
>>> eps = M.character()
>>> R = M.manin_gens_to_basis()
>>> sage.modular.modsym.heilbronn.hecke_images_nonquad_character_weight2(Integer(1),Integer(0),Integer(13),[Integer(1),Integer(2),Integer(6)],eps,R)
[           1            0            0            0]
[   zeta6 + 2            0            0           -1]
[           7 -2*zeta6 + 1   -zeta6 - 1     -2*zeta6]
>>> x = M((Integer(1),Integer(0))); x.element()
(1, 0, 0, 0)
>>> M.T(Integer(2))(x).element()
(zeta6 + 2, 0, 0, -1)
>>> M.T(Integer(6))(x).element()
(7, -2*zeta6 + 1, -zeta6 - 1, -2*zeta6)
sage.modular.modsym.heilbronn.hecke_images_quad_character_weight2(u, v, N, indices, chi, R)[source]#

INPUT:

  • u, v, N – integers so that \(\gcd(u,v,N) = 1\)

  • indices – a list of positive integers

  • chi – a Dirichlet character that takes values in \(\QQ\)

  • R – matrix over \(\QQ(\chi)\) that writes each elements of \(\textnormal{P1} = \textnormal{P1List}(N)\) in terms of a subset of \(\textnormal{P1}\).

OUTPUT: a dense matrix with entries in the rational field \(\QQ\) (the values of \(\chi\)) whose columns are the images \(T_n(x)\) for \(n\) in indices and \(x\) the Manin symbol \((u,v)\), expressed in terms of the basis.

EXAMPLES:

sage: chi = DirichletGroup(29,QQ).0
sage: M = ModularSymbols(chi)
sage: R = M.manin_gens_to_basis()
sage: sage.modular.modsym.heilbronn.hecke_images_quad_character_weight2(2,1,29,[1,3,4],chi,R)
[ 0  0  0  0  0 -1]
[ 0  1  0  1  1  1]
[ 0 -2  0  2 -2 -1]
sage: x = M((2,1)) ; x.element()
(0, 0, 0, 0, 0, -1)
sage: M.T(3)(x).element()
(0, 1, 0, 1, 1, 1)
sage: M.T(4)(x).element()
(0, -2, 0, 2, -2, -1)

>>> from sage.all import *
>>> chi = DirichletGroup(Integer(29),QQ).gen(0)
>>> M = ModularSymbols(chi)
>>> R = M.manin_gens_to_basis()
>>> sage.modular.modsym.heilbronn.hecke_images_quad_character_weight2(Integer(2),Integer(1),Integer(29),[Integer(1),Integer(3),Integer(4)],chi,R)
[ 0  0  0  0  0 -1]
[ 0  1  0  1  1  1]
[ 0 -2  0  2 -2 -1]
>>> x = M((Integer(2),Integer(1))) ; x.element()
(0, 0, 0, 0, 0, -1)
>>> M.T(Integer(3))(x).element()
(0, 1, 0, 1, 1, 1)
>>> M.T(Integer(4))(x).element()
(0, -2, 0, 2, -2, -1)