Module: sage.modular.modsym.manin_symbols
Module-level Functions
i, j) |
i, j, a, b, c, d) |
Returns a list of the coefficients of
where
One should think of
as being
for the application to
modular symbols.
INPUT: i, j, a, b, c, d -- all ints OUTPUT: list of ints, which are the coefficients of Y^j, Y^(j-1)*X, ..., X^j, respectively.
We compute that
.
sage: from sage.modular.modsym.manin_symbols import apply_to_monomial sage: apply_to_monomial(2, 3, 1,1,1,-1) [-1, -1, 1, 1] sage: apply_to_monomial(5, 8, 1,2,3,4) [2048, 9728, 20096, 23584, 17200, 7984, 2304, 378, 27] sage: apply_to_monomial(6,12, 1,1,1,-1) [1, 0, -6, 0, 15, 0, -20, 0, 15, 0, -6, 0, 1]
x) |
Class: ManinSymbol
self, parent, t) |
Create a Manin symbol
, where
is
the weight.
INPUT: parent -- ManinSymbolList t -- a 3-tuple (i,u,v) of int's.
Functions: apply,
copy,
endpoints,
i,
lift_to_sl2z,
modular_symbol_rep,
parent,
tuple,
u,
v,
weight
self, a, b, c, d) |
Return the image of self under the matrix [a,b;c,d].
INPUT: a, b, c, d -- integers OUTPUT: a list of pairs (alpha_i, x_i), where each alpha_i is an integer, x_i is a Manin symbol, and the sum alpha_i*x_i is the image of self under the right action of the matrix [a,b;c,d]. Here the right action of g=[a,b;c,d] on a Manin symbol [P(X,Y),(u,v)] is [P(aX+bY,cX+dY),(u,v)*g].
self, [N=None]) |
Returns cusps alpha, beta such that this Manin symbol, viewed
as a symbol for level N, is
.
self, N) |
If this Manin symbol is (c,d) viewed modulo N, this function computes and returns a list [a,b, c',d'] that defines a 2x2 matrix with determinant 1 and integer entries, such that c=c'(mod N) and d=d'(mod N).
self) |
Returns a representation of self as a formal sum of modular symbols. (The result is not cached.)
Special Functions: __cmp__,
__mul__,
__repr__,
_ManinSymbol__get_i,
_ManinSymbol__get_u,
_ManinSymbol__get_v
Class: ManinSymbolList
self, weight, list) |
Functions: apply,
apply_I,
apply_S,
apply_T,
apply_TT,
index,
manin_symbol,
manin_symbol_list,
normalize,
weight
self, x) |
Return the index into the list of Manin symbols of x, where x is a 3-tuple of ints. If x is not in the list, then this function returns -1.
INPUT: x -- 3-tuple of ints. Something equivalent to an element of Manin symbols list, which need not be normalized. OUTPUT: int -- the index of the Manin symbol equivalent to (i,u,v).
Special Functions: __getitem__,
__len__
Class: ManinSymbolList_character
ManinSymbolList_character(character, weight): INPUT: character - a dirichlet character weight - integer weight >= 2
sage: eps = DirichletGroup(4).gen(0) sage: from sage.modular.modsym.manin_symbols import ManinSymbolList_character sage: m = ManinSymbolList_character(eps,2); m Manin Symbol List of weight 2 for Gamma1(4) with character [-1] sage: m.manin_symbol_list() [(0,1), (1,0), (1,1), (1,2), (1,3), (2,1)]
self, character, weight) |
Functions: apply,
apply_I,
apply_S,
apply_T,
apply_TT,
character,
index,
level,
normalize
self, j, m) |
Apply the matrix m=[a,b,c,d] to the j-th Manin symbol.
INPUT: j -- integer m = [a, b, c, d] a list of 4 integers. OUTPUT: a list of pairs (j, alpha_i), where each alpha_i is an integer, j is an integer (the j-th Manin symbol), and the sum alpha_i*x_i is the image of self under the right action of the matrix [a,b;c,d]. Here the right action of g=[a,b;c,d] on a Manin symbol [P(X,Y),(u,v)] is [P(aX+bY,cX+dY),(u,v)*g].
sage: eps = DirichletGroup(4).gen(0) sage: from sage.modular.modsym.manin_symbols import ManinSymbolList_character sage: m = ManinSymbolList_character(eps,4) sage: m[6] (1, 0, 1) sage: m.apply(4, [1,0,0,1]) [(4, 1)] sage: m.apply(1, [-1,0,0,1]) [(1, -1)]
self, x) |
Compute the index into the list of standard Manin symbols of a symbol that is equivalent, modulo a scalar s, to x. Returns the index and the scalar.
If x is not in the list, then this function returns -1, 0.
INPUT: x -- 3-tuple of ints. Something equivalent to an element of Manin symbols list, which need not be normalized. OUTPUT: int -- the index of the Manin symbol equivalent to (i,u,v). scalar -- element of the base field or the int 0.
Special Functions: __repr__
Class: ManinSymbolList_gamma0
ManinSymbolList_gamma0(level, weight): INPUT: level - integer level weight - integer weight >= 2
sage: from sage.modular.modsym.manin_symbols import ManinSymbolList_gamma0 sage: m = ManinSymbolList_gamma0(5,2); m Manin Symbol List of weight 2 for Gamma0(5) sage: m.manin_symbol_list() [(0,1), (1,0), (1,1), (1,2), (1,3), (1,4)] sage: m = ManinSymbolList_gamma0(6,4); m Manin Symbol List of weight 4 for Gamma0(6) sage: len(m) 36
self, level, weight) |
Functions: apply,
apply_I,
apply_S,
apply_T,
apply_TT,
level,
normalize
self, j, m) |
Apply the matrix m=[a,b,c,d] to the j-th Manin symbol.
INPUT: j -- integer m = [a, b, c, d] a list of 4 integers, which defines a 2x2 matrix. OUTPUT: list -- a list of pairs (j_i, alpha_i), where each alpha_i is a nonzero integer, j_i is an integer (the j_i-th Manin symbol), and the sum alpha_i*x_{j_i} is the image of the j-th Manin symbol under the right action of the matrix [a,b;c,d]. Here the right action of g=[a,b;c,d] on a Manin symbol [P(X,Y),(u,v)] is [P(aX+bY,cX+dY),(u,v)*g].
Special Functions: __repr__
Class: ManinSymbolList_gamma1
sage: from sage.modular.modsym.manin_symbols import ManinSymbolList_gamma0 sage: m = ManinSymbolList_gamma0(5,2); m Manin Symbol List of weight 2 for Gamma0(5) sage: m.manin_symbol_list() [(0,1), (1,0), (1,1), (1,2), (1,3), (1,4)] sage: m = ManinSymbolList_gamma0(6,4); m Manin Symbol List of weight 4 for Gamma0(6) sage: len(m) 36
self, level, weight) |
Functions: apply,
apply_I,
apply_J,
apply_S,
apply_T,
apply_TT,
level,
normalize
self, j, m) |
Apply the matrix m=[a,b,c,d] to the j-th Manin symbol.
INPUT: j -- integer m = [a, b, c, d] a list of 4 integers. OUTPUT: a list of pairs (j, alpha_i), where each alpha_i is an integer, j is an integer (the j-th Manin symbol), and the sum alpha_i*x_i is the image of self under the right action of the matrix [a,b;c,d]. Here the right action of g=[a,b;c,d] on a Manin symbol [P(X,Y),(u,v)] is [P(aX+bY,cX+dY),(u,v)*g].
self, j) |
Apply 2x2 matrix J = [-1,0,0,-1].
Special Functions: __repr__
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