Module: sage.modular.modsym.manin_symbols
Manin symbols
Module-level Functions
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:
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:
Functions: apply,
copy,
endpoints,
lift_to_sl2z,
modular_symbol_rep,
parent,
tuple,
weight
self, a, b, c, d) |
Return the image of self under the matrix [a,b;c,d].
Input:
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__,
__init__,
__mul__,
_latex_,
_ManinSymbol__get_i,
_ManinSymbol__get_u,
_ManinSymbol__get_v,
_repr_
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:
Special Functions: __cmp__,
__getitem__,
__init__,
__len__
Class: ManinSymbolList_character
ManinSymbolList_character(character, weight): Input:
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:
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:
Special Functions: __init__,
__repr__
Class: ManinSymbolList_gamma0
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) |
Special Functions: __init__,
__repr__
Class: ManinSymbolList_gamma1
self, level, weight) |
Special Functions: __init__,
__repr__
Class: ManinSymbolList_gamma_h
self, group, weight) |
Special Functions: __init__,
__repr__
Class: ManinSymbolList_group
ManinSymbolList_group(level, weight, syms):
Input:
self, level, weight, syms) |
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:
Special Functions: __init__
Class: x__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:
self, j) |
Apply 2x2 matrix J = [-1,0,0,-1].
Special Functions: __init__,
__repr__
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