# Manin symbol lists¶

There are various different classes holding lists of Manin symbols of different types. The hierarchy is as follows:

class sage.modular.modsym.manin_symbol_list.ManinSymbolList(weight, lst)

Base class for lists of all Manin symbols for a given weight, group or character.

Element
apply(j, X)

Apply the matrix $$X = [a, b; c, d]$$ to the $$j$$-th Manin symbol.

Implemented in derived classes.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.apply(10, [1,2,0,1])
Traceback (most recent call last):
...
NotImplementedError: Only implemented in derived classes

apply_I(j)

Apply the matrix $$I = [-1, 0; 0, 1]$$ to the $$j$$-th Manin symbol.

Implemented in derived classes.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.apply_I(10)
Traceback (most recent call last):
...
NotImplementedError: Only implemented in derived classes

apply_S(j)

Apply the matrix $$S = [0, -1; 1, 0]$$ to the $$j$$-th Manin symbol.

Implemented in derived classes.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.apply_S(10)
Traceback (most recent call last):
...
NotImplementedError: Only implemented in derived classes

apply_T(j)

Apply the matrix $$T = [0, 1; -1, -1]$$ to the $$j$$-th Manin symbol.

Implemented in derived classes.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.apply_T(10)
Traceback (most recent call last):
...
NotImplementedError: Only implemented in derived classes

apply_TT(j)

Apply the matrix $$TT = T^2 = [-1, -1; 0, 1]$$ to the $$j$$-th Manin symbol.

Implemented in derived classes.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.apply_TT(10)
Traceback (most recent call last):
...
NotImplementedError: Only implemented in derived classes

index(x)

Return the index of x in the list of Manin symbols.

INPUT:

• x – a triple of integers $$(i, u, v)$$ defining a valid Manin symbol, which need not be normalized

OUTPUT:

integer – the index of the normalized Manin symbol equivalent to $$(i, u, v)$$. If x is not in self, -1 is returned.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.index(m.symbol_list()[2])
2
sage: S = m.symbol_list()
sage: all(i == m.index(S[i]) for i in range(len(S)))
True

list()

Return all the Manin symbols in self as a list.

Cached for subsequent calls.

OUTPUT:

A list of ManinSymbol objects, which is a copy of the complete list of Manin symbols.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.manin_symbol_list() # not implemented for the base class

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(6, 4)
sage: m.manin_symbol_list()
[[Y^2,(0,1)],
[Y^2,(1,0)],
[Y^2,(1,1)],
...
[X^2,(3,1)],
[X^2,(3,2)]]

manin_symbol(i)

Return the i-th Manin symbol in this ManinSymbolList.

INPUT:

• i – integer, a valid index of a symbol in this list

OUTPUT:

ManinSymbol – the $$i$$’th Manin symbol in the list.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.manin_symbol(3) # not implemented for base class

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(6, 4)
sage: s = m.manin_symbol(3); s
[Y^2,(1,2)]
sage: type(s)
<type 'sage.modular.modsym.manin_symbol.ManinSymbol'>

manin_symbol_list()

Return all the Manin symbols in self as a list.

Cached for subsequent calls.

OUTPUT:

A list of ManinSymbol objects, which is a copy of the complete list of Manin symbols.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.manin_symbol_list() # not implemented for the base class

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(6, 4)
sage: m.manin_symbol_list()
[[Y^2,(0,1)],
[Y^2,(1,0)],
[Y^2,(1,1)],
...
[X^2,(3,1)],
[X^2,(3,2)]]

normalize(x)

Return a normalized Manin symbol from x.

To be implemented in derived classes.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6,P1List(11))
sage: m.normalize((0,6,7)) # not implemented in base class

symbol_list()

Return the list of symbols of self.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList
sage: m = ManinSymbolList(6, P1List(11))

weight()

Return the weight of the Manin symbols in this ManinSymbolList.

OUTPUT:

integer – the weight of the Manin symbols in the list.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(6, 4)
sage: m.weight()
4

class sage.modular.modsym.manin_symbol_list.ManinSymbolList_character(character, weight)

List of Manin symbols with character.

INPUT:

• character – (DirichletCharacter) the Dirichlet character
• weight – (integer) the weight

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list 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)]
True

apply(j, m)

Apply the integer matrix $$m=[a,b;c,d]$$ to the $$j$$-th Manin symbol.

INPUT:

• j (integer): the index of the symbol to act on.
• m (list of ints): $$[a,b,c,d]$$ where $$m = [a, b; c, d]$$ is the matrix to be applied.

OUTPUT:

A list of pairs $$(j, c_i)$$, where each $$c_i$$ is an integer, $$j$$ is an integer (the $$j$$-th Manin symbol), and the sum $$c_i*x_i$$ 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 by definition $$[P(aX+bY,cX+dY),(u,v)*g]$$.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,4)
sage: m[6]
[X*Y,(0,1)]
sage: m.apply(4, [1,0,0,1])
[(4, 1)]
sage: m.apply(1, [-1,0,0,1])
[(1, -1)]

apply_I(j)

Apply the matrix $$I=[-1,0,0,1]$$ to the $$j$$-th Manin symbol.

INPUT:

• j - (integer) a symbol index

OUTPUT:

(k, s) where $$k$$ is the index of the symbol obtained by acting on the $$j$$’th symbol with $$I$$, and $$s$$ is the parity of the $$j$$’th symbol.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,2); m
Manin Symbol List of weight 2 for Gamma1(4) with character [-1]
sage: m.apply_I(4)
(2, -1)
sage: [m.apply_I(i) for i in range(len(m))]
[(0, 1), (1, -1), (4, -1), (3, -1), (2, -1), (5, 1)]

apply_S(j)

Apply the matrix $$S=[0,1;-1,0]$$ to the $$j$$-th Manin symbol.

INPUT:

• j - (integer) a symbol index.

OUTPUT:

(k, s) where $$k$$ is the index of the symbol obtained by acting on the $$j$$’th symbol with $$S$$, and $$s$$ is the parity of the $$j$$’th symbol.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,2); m
Manin Symbol List of weight 2 for Gamma1(4) with character [-1]
sage: m.apply_S(4)
(2, -1)
sage: [m.apply_S(i) for i in range(len(m))]
[(1, 1), (0, -1), (4, 1), (5, -1), (2, -1), (3, 1)]

apply_T(j)

Apply the matrix $$T=[0,1,-1,-1]$$ to the j-th Manin symbol.

INPUT:

• j - (integer) a symbol index.

OUTPUT:

A list of pairs $$(j, c_i)$$, where each $$c_i$$ is an integer, $$j$$ is an integer (the $$j$$-th Manin symbol), and the sum $$c_i*x_i$$ is the image of self under the right action of the matrix $$T$$.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,2); m
Manin Symbol List of weight 2 for Gamma1(4) with character [-1]
sage: m.apply_T(4)
[(1, -1)]
sage: [m.apply_T(i) for i in range(len(m))]
[[(4, 1)], [(0, -1)], [(3, 1)], [(5, 1)], [(1, -1)], [(2, 1)]]

apply_TT(j)

Apply the matrix $$TT=[-1,-1,0,1]$$ to the $$j$$-th Manin symbol.

INPUT:

• j - (integer) a symbol index

OUTPUT:

A list of pairs $$(j, c_i)$$, where each $$c_i$$ is an integer, $$j$$ is an integer (the $$j$$-th Manin symbol), and the sum $$c_i*x_i$$ is the image of self under the right action of the matrix $$T^2$$.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,2); m
Manin Symbol List of weight 2 for Gamma1(4) with character [-1]
sage: m.apply_TT(4)
[(0, 1)]
sage: [m.apply_TT(i) for i in range(len(m))]
[[(1, -1)], [(4, -1)], [(5, 1)], [(2, 1)], [(0, 1)], [(3, 1)]]

character()

Return the character of this ManinSymbolList_character object.

OUTPUT:

The Dirichlet character of this Manin symbol list.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,2); m
Manin Symbol List of weight 2 for Gamma1(4) with character [-1]
sage: m.character()
Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1

index(x)

Return the index of a standard Manin symbol equivalent to x, together with a scaling factor.

INPUT:

• x – 3-tuple of integers defining an element of this list of Manin symbols, which need not be normalized

OUTPUT:

A pair (i, s) where i is the index of the Manin symbol equivalent to x and s is the scalar (an element of the base field). If there is no Manin symbol equivalent to x in the list, then (-1, 0) is returned.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,4); m
Manin Symbol List of weight 4 for Gamma1(4) with character [-1]
sage: [m.index(s.tuple()) for s in m.manin_symbol_list()]
[(0, 1),
(1, 1),
(2, 1),
(3, 1),
...
(16, 1),
(17, 1)]

level()

Return the level of this ManinSymbolList.

OUTPUT:

integer - the level of the symbols in this list.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: ManinSymbolList_character(eps,4).level()
4

normalize(x)

Return the normalization of the Manin Symbol x with respect to this list, together with the normalizing scalar.

INPUT:

• x - 3-tuple of integers (i,u,v), defining an element of this list of Manin symbols, which need not be normalized.

OUTPUT:

((i,u,v),s), where (i,u,v) is the normalized Manin symbol equivalent to x, and s is the normalizing scalar.

EXAMPLES:

sage: eps = DirichletGroup(4).gen(0)
sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_character
sage: m = ManinSymbolList_character(eps,4); m
Manin Symbol List of weight 4 for Gamma1(4) with character [-1]
sage: [m.normalize(s.tuple()) for s in m.manin_symbol_list()]
[((0, 0, 1), 1),
((0, 1, 0), 1),
((0, 1, 1), 1),
...
((2, 1, 3), 1),
((2, 2, 1), 1)]

class sage.modular.modsym.manin_symbol_list.ManinSymbolList_gamma0(level, weight)

Class for Manin symbols for $$\Gamma_0(N)$$.

INPUT:

• level - (integer): the level.
• weight - (integer): the weight.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list 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

class sage.modular.modsym.manin_symbol_list.ManinSymbolList_gamma1(level, weight)

Class for Manin symbols for $$\Gamma_1(N)$$.

INPUT:

• level - (integer): the level.
• weight - (integer): the weight.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma1
sage: m = ManinSymbolList_gamma1(5,2); m
Manin Symbol List of weight 2 for Gamma1(5)
sage: m.manin_symbol_list()
[(0,1),
(0,2),
(0,3),
...
(4,3),
(4,4)]
sage: m = ManinSymbolList_gamma1(6,4); m
Manin Symbol List of weight 4 for Gamma1(6)
sage: len(m)
72
True

class sage.modular.modsym.manin_symbol_list.ManinSymbolList_gamma_h(group, weight)

Class for Manin symbols for $$\Gamma_H(N)$$.

INPUT:

• group - (integer): the congruence subgroup.
• weight - (integer): the weight.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma_h
sage: G = GammaH(117, [4])
sage: m = ManinSymbolList_gamma_h(G,2); m
Manin Symbol List of weight 2 for Congruence Subgroup Gamma_H(117) with H generated by [4]
sage: m.manin_symbol_list()[100:110]
[(1,88),
(1,89),
(1,90),
(1,91),
(1,92),
(1,93),
(1,94),
(1,95),
(1,96),
(1,97)]
sage: len(m.manin_symbol_list())
2016
True

group()

Return the group associated to self.

EXAMPLES:

sage: ModularSymbols(GammaH(12, [5]), 2).manin_symbols().group()
Congruence Subgroup Gamma_H(12) with H generated by [5]

class sage.modular.modsym.manin_symbol_list.ManinSymbolList_group(level, weight, syms)

Base class for Manin symbol lists for a given group.

INPUT:

• level – integer level
• weight – integer weight
• syms – something with normalize and list methods,
e.g. P1List.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_group
sage: ManinSymbolList_group(11, 2, P1List(11))
<sage.modular.modsym.manin_symbol_list.ManinSymbolList_group_with_category object at ...>

apply(j, m)

Apply the matrix $$m = [a, b; c, d]$$ to the $$j$$-th Manin symbol.

INPUT:

• j - (int) a symbol index
• m = [a, b, c, d] a list of 4 integers, which defines a 2x2 matrix

OUTPUT:

a list of pairs $$(j_i, \alpha_i)$$, where each $$\alpha_i$$ is a nonzero integer, $$j_i$$ is an integer (index of the $$j_i$$-th Manin symbol), and $$\sum_i \alpha_i\*x_{j_i}$$ 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 $$[P(aX+bY,cX+dY),(u,v)\*g]$$.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply(40, [2,3,1,1])
[(0, 729), (6, 2916), (12, 4860), (18, 4320),
(24, 2160), (30, 576), (36, 64)]

apply_I(j)

Apply the matrix $$I=[-1,0,0,1]$$ to the $$j$$-th Manin symbol.

INPUT:

• j - (int) a symbol index

OUTPUT:

(k, s) where k is the index of the symbol obtained by acting on the $$j$$’th symbol with $$I$$, and $$s$$ is the parity of the $$j$$’th symbol (a Python int, either 1 or -1)

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply_I(4)
(3, 1)
sage: [m.apply_I(i) for i in range(10)]
[(0, 1),
(1, 1),
(5, 1),
(4, 1),
(3, 1),
(2, 1),
(6, -1),
(7, -1),
(11, -1),
(10, -1)]

apply_S(j)

Apply the matrix $$S = [0, -1; 1, 0]$$ to the $$j$$-th Manin symbol.

INPUT:

• j – (int) a symbol index

OUTPUT:

(k, s) where k is the index of the symbol obtained by acting on the $$j$$’th symbol with $$S$$, and $$s$$ is the parity of the $$j$$’th symbol (a Python int, either 1 or -1).

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply_S(4)
(40, 1)
sage: [m.apply_S(i) for i in range(len(m))]
[(37, 1),
(36, 1),
(41, 1),
(39, 1),
(40, 1),
(38, 1),
(31, -1),
(30, -1),
(35, -1),
(33, -1),
(34, -1),
(32, -1),
...
(4, 1),
(2, 1)]

apply_T(j)

Apply the matrix $$T=[0,1,-1,-1]$$ to the $$j$$-th Manin symbol.

INPUT:

• j - (int) a symbol index

OUTPUT: see documentation for apply()

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply_T(4)
[(3, 1), (9, -6), (15, 15), (21, -20), (27, 15), (33, -6), (39, 1)]
sage: [m.apply_T(i) for i in range(10)]
[[(5, 1), (11, -6), (17, 15), (23, -20), (29, 15), (35, -6), (41, 1)],
[(0, 1), (6, -6), (12, 15), (18, -20), (24, 15), (30, -6), (36, 1)],
[(4, 1), (10, -6), (16, 15), (22, -20), (28, 15), (34, -6), (40, 1)],
[(2, 1), (8, -6), (14, 15), (20, -20), (26, 15), (32, -6), (38, 1)],
[(3, 1), (9, -6), (15, 15), (21, -20), (27, 15), (33, -6), (39, 1)],
[(1, 1), (7, -6), (13, 15), (19, -20), (25, 15), (31, -6), (37, 1)],
[(5, 1), (11, -5), (17, 10), (23, -10), (29, 5), (35, -1)],
[(0, 1), (6, -5), (12, 10), (18, -10), (24, 5), (30, -1)],
[(4, 1), (10, -5), (16, 10), (22, -10), (28, 5), (34, -1)],
[(2, 1), (8, -5), (14, 10), (20, -10), (26, 5), (32, -1)]]

apply_TT(j)

Apply the matrix $$TT=[-1,-1,0,1]$$ to the $$j$$-th Manin symbol.

INPUT:

• j - (int) a symbol index

OUTPUT: see documentation for apply()

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: m.apply_TT(4)
[(38, 1)]
sage: [m.apply_TT(i) for i in range(10)]
[[(37, 1)],
[(41, 1)],
[(39, 1)],
[(40, 1)],
[(38, 1)],
[(36, 1)],
[(31, -1), (37, 1)],
[(35, -1), (41, 1)],
[(33, -1), (39, 1)],
[(34, -1), (40, 1)]]

level()

Return the level of this ManinSymbolList.

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: ManinSymbolList_gamma0(5,2).level()
5

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma1
sage: ManinSymbolList_gamma1(51,2).level()
51

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma_h
sage: ManinSymbolList_gamma_h(GammaH(117, [4]),2).level()
117

normalize(x)

Return the normalization of the Manin symbol x with respect to this list.

INPUT:

• x – (3-tuple of ints) a tuple defining a ManinSymbol

OUTPUT:

(i,u,v) – (3-tuple of ints) another tuple defining the associated normalized ManinSymbol

EXAMPLES:

sage: from sage.modular.modsym.manin_symbol_list import ManinSymbolList_gamma0
sage: m = ManinSymbolList_gamma0(5,8)
sage: [m.normalize(s.tuple()) for s in m.manin_symbol_list()][:10]
[(0, 0, 1),
(0, 1, 0),
(0, 1, 1),
(0, 1, 2),
(0, 1, 3),
(0, 1, 4),
(1, 0, 1),
(1, 1, 0),
(1, 1, 1),
(1, 1, 2)]