8.6.2 modular.modsym.manin_symbols - Manin symbols

The module modular.modsym.manin_symbols defines the following classes:

class ManinSymbol
A Manin symbol $ [X^i\cdot Y^{k-2-i},(u,v)]$ .

class ManinSymbolList
All Manin symbols for a given group, weight, and character.

class ManinSymbolList_character
List of Manin Symbols with character.

ManinSymbolList_character(character, weight): INPUT: character - a dirichlet character weight - integer weight >= 2 EXAMPLE:

sage: eps = DirichletGroup(4).gen(0)
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)]

class ManinSymbolList_gamma0
List of Manin symbols for Gamma0(N).

ManinSymbolList_gamma0(level, weight): INPUT: level - integer level weight - integer weight >= 2

EXAMPLE:

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

class ManinSymbolList_gamma1
List of Manin symbols for Gamma0(N).

EXAMPLE:

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

The module modular.modsym.manin_symbols defines the following functions:

apply_to_monomial( i, j, a, b, c, d)

Returns a list of the coefficients of

$\displaystyle (aX + bY)^i (cX + dY)^{j-i},
$

where $ 0 \leq i \leq j$ , and $ a,b,c,d$ are integers.

One should think of $ j$ as being $ k-2$ 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.

EXAMPLE: We compute that $ (X+Y)^2(X-Y) = X^3 + X^2Y - XY^2 - Y^3$ .

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]



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