Dirichlet characters

A DirichletCharacter is the extension of a homomorphism

\[(\ZZ/N\ZZ)^* \to R^*,\]

for some ring \(R\), to the map \(\ZZ/N\ZZ \to R\) obtained by sending those \(x\in\ZZ/N\ZZ\) with \(\gcd(N,x)>1\) to \(0\).

EXAMPLES:

sage: G = DirichletGroup(35)
sage: x = G.gens()
sage: e = x[0]*x[1]^2; e
Dirichlet character modulo 35 of conductor 35 mapping 22 |--> zeta12^3, 31 |--> zeta12^2 - 1
sage: e.order()
12

This illustrates a canonical coercion:

sage: e = DirichletGroup(5, QQ).0
sage: f = DirichletGroup(5,CyclotomicField(4)).0
sage: e*f
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4

AUTHORS:

  • William Stein (2005-09-02): Fixed bug in comparison of Dirichlet characters. It was checking that their values were the same, but not checking that they had the same level!
  • William Stein (2006-01-07): added more examples
  • William Stein (2006-05-21): added examples of everything; fix a lot of tiny bugs and design problem that became clear when creating examples.
  • Craig Citro (2008-02-16): speed up __call__ method for Dirichlet characters, miscellaneous fixes
  • Julian Rueth (2014-03-06): use UniqueFactory to cache DirichletGroups
class sage.modular.dirichlet.DirichletCharacter(parent, x, check=True)

Bases: sage.structure.element.MultiplicativeGroupElement

A Dirichlet character.

bar()

Return the complex conjugate of this Dirichlet character.

EXAMPLES:

sage: e = DirichletGroup(5).0
sage: e
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4
sage: e.bar()
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4
base_ring()

Returns the base ring of this Dirichlet character.

EXAMPLES:

sage: G = DirichletGroup(11)
sage: G.gen(0).base_ring()
Cyclotomic Field of order 10 and degree 4
sage: G = DirichletGroup(11, RationalField())
sage: G.gen(0).base_ring()
Rational Field
bernoulli(k, algorithm='recurrence', cache=True, **opts)

Returns the generalized Bernoulli number \(B_{k,eps}\).

INPUT:

  • k – a non-negative integer
  • algorithm – either 'recurrence' (default) or 'definition'
  • cache – if True, cache answers
  • **opts – optional arguments; not used directly, but passed to the bernoulli() function if this is called

OUTPUT:

Let \(\varepsilon\) be a (not necessarily primitive) character of modulus \(N\). This function returns the generalized Bernoulli number \(B_{k,\varepsilon}\), as defined by the following identity of power series (see for example [DI1995], Section 2.2):

\[\sum_{a=1}^N \frac{\varepsilon(a) t e^{at}}{e^{Nt}-1} = sum_{k=0}^{\infty} \frac{B_{k,\varepsilon}}{k!} t^k.\]

ALGORITHM:

The 'recurrence' algorithm computes generalized Bernoulli numbers via classical Bernoulli numbers using the formula in [Coh2007], Proposition 9.4.5; this is usually optimal. The definition algorithm uses the definition directly.

Warning

In the case of the trivial Dirichlet character modulo 1, this function returns \(B_{1,\varepsilon} = 1/2\), in accordance with the above definition, but in contrast to the value \(B_1 = -1/2\) for the classical Bernoulli number. Some authors use an alternative definition giving \(B_{1,\varepsilon} = -1/2\); see the discussion in [Coh2007], Section 9.4.1.

EXAMPLES:

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.bernoulli(5)
7430/13*zeta12^3 - 34750/13*zeta12^2 - 11380/13*zeta12 + 9110/13
sage: eps = DirichletGroup(9).0
sage: eps.bernoulli(3)
10*zeta6 + 4
sage: eps.bernoulli(3, algorithm="definition")
10*zeta6 + 4
change_ring(R)

Return the base extension of self to R.

INPUT:

  • R – either a ring admitting a conversion map from the base ring of self, or a ring homomorphism with the base ring of self as its domain

EXAMPLES:

sage: e = DirichletGroup(7, QQ).0
sage: f = e.change_ring(QuadraticField(3, 'a'))
sage: f.parent()
Group of Dirichlet characters modulo 7 with values in Number Field in a with defining polynomial x^2 - 3
sage: e = DirichletGroup(13).0
sage: e.change_ring(QQ)
Traceback (most recent call last):
...
TypeError: Unable to coerce zeta12 to a rational

We test the case where \(R\) is a map (trac ticket #18072):

sage: K.<i> = QuadraticField(-1)
sage: chi = DirichletGroup(5, K)[1]
sage: chi(2)
i
sage: f = K.complex_embeddings()[0]
sage: psi = chi.change_ring(f)
sage: psi(2)
-1.83697019872103e-16 - 1.00000000000000*I
conductor()

Computes and returns the conductor of this character.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: a.conductor()
4
sage: b.conductor()
5
sage: (a*b).conductor()
20
decomposition()

Return the decomposition of self as a product of Dirichlet characters of prime power modulus, where the prime powers exactly divide the modulus of this character.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: c = a*b
sage: d = c.decomposition(); d
[Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1, Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4]
sage: d[0].parent()
Group of Dirichlet characters modulo 4 with values in Cyclotomic Field of order 4 and degree 2
sage: d[1].parent()
Group of Dirichlet characters modulo 5 with values in Cyclotomic Field of order 4 and degree 2

We can’t multiply directly, since coercion of one element into the other parent fails in both cases:

sage: d[0]*d[1] == c
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *: 'Group of Dirichlet characters modulo 4 with values in Cyclotomic Field of order 4 and degree 2' and 'Group of Dirichlet characters modulo 5 with values in Cyclotomic Field of order 4 and degree 2'

We can multiply if we’re explicit about where we want the multiplication to take place.

sage: G(d[0])*G(d[1]) == c
True

Conductors that are divisible by various powers of 2 present some problems as the multiplicative group modulo \(2^k\) is trivial for \(k = 1\) and non-cyclic for \(k \ge 3\):

sage: (DirichletGroup(18).0).decomposition()
[Dirichlet character modulo 2 of conductor 1, Dirichlet character modulo 9 of conductor 9 mapping 2 |--> zeta6]
sage: (DirichletGroup(36).0).decomposition()
[Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1, Dirichlet character modulo 9 of conductor 1 mapping 2 |--> 1]
sage: (DirichletGroup(72).0).decomposition()
[Dirichlet character modulo 8 of conductor 4 mapping 7 |--> -1, 5 |--> 1, Dirichlet character modulo 9 of conductor 1 mapping 2 |--> 1]
element()

Return the underlying \(\ZZ/n\ZZ\)-module vector of exponents.

Warning

Please do not change the entries of the returned vector; this vector is mutable only because immutable vectors are not implemented yet.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: a.element()
(2, 0)
sage: b.element()
(0, 1)

Note

The constructor of DirichletCharacter sets the cache of element() or of values_on_gens(). The cache of one of these methods needs to be set for the other method to work properly, these caches have to be stored when pickling an instance of DirichletCharacter.

extend(M)

Returns the extension of this character to a Dirichlet character modulo the multiple M of the modulus.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: H.<c> = DirichletGroup(4)
sage: c.extend(20)
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: a
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: c.extend(20) == a
True
galois_orbit(sort=True)

Return the orbit of this character under the action of the absolute Galois group of the prime subfield of the base ring.

EXAMPLES:

sage: G = DirichletGroup(30); e = G.1
sage: e.galois_orbit()
[Dirichlet character modulo 30 of conductor 5 mapping 11 |--> 1, 7 |--> -zeta4,
 Dirichlet character modulo 30 of conductor 5 mapping 11 |--> 1, 7 |--> zeta4]

Another example:

sage: G = DirichletGroup(13)
sage: G.galois_orbits()
[
[Dirichlet character modulo 13 of conductor 1 mapping 2 |--> 1],
...,
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -1]
]
sage: e = G.0
sage: e
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12
sage: e.galois_orbit()
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12^3 + zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^3 - zeta12,
 Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12]
sage: e = G.0^2; e
Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^2
sage: e.galois_orbit()
[Dirichlet character modulo 13 of conductor 13 mapping 2 |--> zeta12^2, Dirichlet character modulo 13 of conductor 13 mapping 2 |--> -zeta12^2 + 1]

A non-example:

sage: chi = DirichletGroup(7, Integers(9), zeta = Integers(9)(2)).0
sage: chi.galois_orbit()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain
gauss_sum(a=1)

Return a Gauss sum associated to this Dirichlet character.

The Gauss sum associated to \(\chi\) is

\[g_a(\chi) = \sum_{r \in \ZZ/m\ZZ} \chi(r)\,\zeta^{ar},\]

where \(m\) is the modulus of \(\chi\) and \(\zeta\) is a primitive \(m^{th}\) root of unity.

FACTS: If the modulus is a prime \(p\) and the character is nontrivial, then the Gauss sum has absolute value \(\sqrt{p}\).

CACHING: Computed Gauss sums are not cached with this character.

EXAMPLES:

sage: G = DirichletGroup(3)
sage: e = G([-1])
sage: e.gauss_sum(1)
2*zeta6 - 1
sage: e.gauss_sum(2)
-2*zeta6 + 1
sage: norm(e.gauss_sum())
3
sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.gauss_sum()
-zeta156^46 + zeta156^45 + zeta156^42 + zeta156^41 + 2*zeta156^40 + zeta156^37 - zeta156^36 - zeta156^34 - zeta156^33 - zeta156^31 + 2*zeta156^30 + zeta156^28 - zeta156^24 - zeta156^22 + zeta156^21 + zeta156^20 - zeta156^19 + zeta156^18 - zeta156^16 - zeta156^15 - 2*zeta156^14 - zeta156^10 + zeta156^8 + zeta156^7 + zeta156^6 + zeta156^5 - zeta156^4 - zeta156^2 - 1
sage: factor(norm(e.gauss_sum()))
13^24

See also

gauss_sum_numerical(prec=53, a=1)

Return a Gauss sum associated to this Dirichlet character as an approximate complex number with prec bits of precision.

INPUT:

  • prec – integer (default: 53), bits of precision
  • a – integer, as for gauss_sum().

The Gauss sum associated to \(\chi\) is

\[g_a(\chi) = \sum_{r \in \ZZ/m\ZZ} \chi(r)\,\zeta^{ar},\]

where \(m\) is the modulus of \(\chi\) and \(\zeta\) is a primitive \(m^{th}\) root of unity.

EXAMPLES:

sage: G = DirichletGroup(3)
sage: e = G.0
sage: abs(e.gauss_sum_numerical())
1.7320508075...
sage: sqrt(3.0)
1.73205080756888
sage: e.gauss_sum_numerical(a=2)
-...e-15 - 1.7320508075...*I
sage: e.gauss_sum_numerical(a=2, prec=100)
4.7331654313260708324703713917e-30 - 1.7320508075688772935274463415*I
sage: G = DirichletGroup(13)
sage: H = DirichletGroup(13, CC)
sage: e = G.0
sage: f = H.0
sage: e.gauss_sum_numerical()
-3.07497205... + 1.8826966926...*I
sage: f.gauss_sum_numerical()
-3.07497205... + 1.8826966926...*I
sage: abs(e.gauss_sum_numerical())
3.60555127546...
sage: abs(f.gauss_sum_numerical())
3.60555127546...
sage: sqrt(13.0)
3.60555127546399
is_even()

Return True if and only if \(\varepsilon(-1) = 1\).

EXAMPLES:

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.is_even()
False
sage: e(-1)
-1
sage: [e.is_even() for e in G]
[True, False, True, False, True, False, True, False, True, False, True, False]

sage: G = DirichletGroup(13, CC)
sage: e = G.0
sage: e.is_even()
False
sage: e(-1)
-1.000000...
sage: [e.is_even() for e in G]
[True, False, True, False, True, False, True, False, True, False, True, False]

sage: G = DirichletGroup(100000, CC)
sage: G.1.is_even()
True

Note that is_even need not be the negation of is_odd, e.g., in characteristic 2:

sage: G.<e> = DirichletGroup(13, GF(4,'a'))
sage: e.is_even()
True
sage: e.is_odd()
True
is_odd()

Return True if and only if \(\varepsilon(-1) = -1\).

EXAMPLES:

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.is_odd()
True
sage: [e.is_odd() for e in G]
[False, True, False, True, False, True, False, True, False, True, False, True]

sage: G = DirichletGroup(13)
sage: e = G.0
sage: e.is_odd()
True
sage: [e.is_odd() for e in G]
[False, True, False, True, False, True, False, True, False, True, False, True]

sage: G = DirichletGroup(100000, CC)
sage: G.0.is_odd()
True

Note that is_even need not be the negation of is_odd, e.g., in characteristic 2:

sage: G.<e> = DirichletGroup(13, GF(4,'a'))
sage: e.is_even()
True
sage: e.is_odd()
True
is_primitive()

Return True if and only if this character is primitive, i.e., its conductor equals its modulus.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: a.is_primitive()
False
sage: b.is_primitive()
False
sage: (a*b).is_primitive()
True
sage: G.<a,b> = DirichletGroup(20, CC)
sage: a.is_primitive()
False
sage: b.is_primitive()
False
sage: (a*b).is_primitive()
True
is_trivial()

Returns True if this is the trivial character, i.e., has order 1.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: a.is_trivial()
False
sage: (a^2).is_trivial()
True
jacobi_sum(char, check=True)

Return the Jacobi sum associated to these Dirichlet characters (i.e., J(self,char)).

This is defined as

\[J(\chi, \psi) = \sum_{a \in \ZZ / N\ZZ} \chi(a) \psi(1-a)\]

where \(\chi\) and \(\psi\) are both characters modulo \(N\).

EXAMPLES:

sage: D = DirichletGroup(13)
sage: e = D.0
sage: f = D[-2]
sage: e.jacobi_sum(f)
3*zeta12^2 + 2*zeta12 - 3
sage: f.jacobi_sum(e)
3*zeta12^2 + 2*zeta12 - 3
sage: p = 7
sage: DP = DirichletGroup(p)
sage: f = DP.0
sage: e.jacobi_sum(f)
Traceback (most recent call last):
...
NotImplementedError: Characters must be from the same Dirichlet Group.

sage: all_jacobi_sums = [(DP[i].values_on_gens(),DP[j].values_on_gens(),DP[i].jacobi_sum(DP[j]))
....:                   for i in range(p-1) for j in range(i, p-1)]
sage: for s in all_jacobi_sums:
....:     print(s)
((1,), (1,), 5)
((1,), (zeta6,), -1)
((1,), (zeta6 - 1,), -1)
((1,), (-1,), -1)
((1,), (-zeta6,), -1)
((1,), (-zeta6 + 1,), -1)
((zeta6,), (zeta6,), -zeta6 + 3)
((zeta6,), (zeta6 - 1,), 2*zeta6 + 1)
((zeta6,), (-1,), -2*zeta6 - 1)
((zeta6,), (-zeta6,), zeta6 - 3)
((zeta6,), (-zeta6 + 1,), 1)
((zeta6 - 1,), (zeta6 - 1,), -3*zeta6 + 2)
((zeta6 - 1,), (-1,), 2*zeta6 + 1)
((zeta6 - 1,), (-zeta6,), -1)
((zeta6 - 1,), (-zeta6 + 1,), -zeta6 - 2)
((-1,), (-1,), 1)
((-1,), (-zeta6,), -2*zeta6 + 3)
((-1,), (-zeta6 + 1,), 2*zeta6 - 3)
((-zeta6,), (-zeta6,), 3*zeta6 - 1)
((-zeta6,), (-zeta6 + 1,), -2*zeta6 + 3)
((-zeta6 + 1,), (-zeta6 + 1,), zeta6 + 2)

Let’s check that trivial sums are being calculated correctly:

sage: N = 13
sage: D = DirichletGroup(N)
sage: g = D(1)
sage: g.jacobi_sum(g)
11
sage: sum([g(x)*g(1-x) for x in IntegerModRing(N)])
11

And sums where exactly one character is nontrivial (see trac ticket #6393):

sage: G = DirichletGroup(5); X=G.list(); Y=X[0]; Z=X[1]
sage: Y.jacobi_sum(Z)
-1
sage: Z.jacobi_sum(Y)
-1

Now let’s take a look at a non-prime modulus:

sage: N = 9
sage: D = DirichletGroup(N)
sage: g = D(1)
sage: g.jacobi_sum(g)
3

We consider a sum with values in a finite field:

sage: g = DirichletGroup(17, GF(9,'a')).0
sage: g.jacobi_sum(g**2)
2*a
kernel()

Return the kernel of this character.

OUTPUT: Currently the kernel is returned as a list. This may change.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: a.kernel()
[1, 9, 13, 17]
sage: b.kernel()
[1, 11]
kloosterman_sum(a=1, b=0)

Return the “twisted” Kloosterman sum associated to this Dirichlet character.

This includes Gauss sums, classical Kloosterman sums, Salié sums, etc.

The Kloosterman sum associated to \(\chi\) and the integers a,b is

\[K(a,b,\chi) = \sum_{r \in (\ZZ/m\ZZ)^\times} \chi(r)\,\zeta^{ar+br^{-1}},\]

where \(m\) is the modulus of \(\chi\) and \(\zeta\) is a primitive \(m\) th root of unity. This reduces to the Gauss sum if \(b=0\).

This method performs an exact calculation and returns an element of a suitable cyclotomic field; see also kloosterman_sum_numerical(), which gives an inexact answer (but is generally much quicker).

CACHING: Computed Kloosterman sums are not cached with this character.

EXAMPLES:

sage: G = DirichletGroup(3)
sage: e = G([-1])
sage: e.kloosterman_sum(3,5)
-2*zeta6 + 1
sage: G = DirichletGroup(20)
sage: e = G([1 for  u in G.unit_gens()])
sage: e.kloosterman_sum(7,17)
-2*zeta20^6 + 2*zeta20^4 + 4
kloosterman_sum_numerical(prec=53, a=1, b=0)

Return the Kloosterman sum associated to this Dirichlet character as an approximate complex number with prec bits of precision. See also kloosterman_sum(), which calculates the sum exactly (which is generally slower).

INPUT:

EXAMPLES:

sage: G = DirichletGroup(3)
sage: e = G.0

The real component of the numerical value of e is near zero:

sage: v=e.kloosterman_sum_numerical()
sage: v.real() < 1.0e15
True
sage: v.imag()
1.73205080756888
sage: G = DirichletGroup(20)
sage: e = G.1
sage: e.kloosterman_sum_numerical(53,3,11)
3.80422606518061 - 3.80422606518061*I
level()

Synonym for modulus.

EXAMPLES:

sage: e = DirichletGroup(100, QQ).0
sage: e.level()
100
maximize_base_ring()

Let

\[\varepsilon : (\ZZ/N\ZZ)^* \to \QQ(\zeta_n)\]

be a Dirichlet character. This function returns an equal Dirichlet character

\[\chi : (\ZZ/N\ZZ)^* \to \QQ(\zeta_m)\]

where \(m\) is the least common multiple of \(n\) and the exponent of \((\ZZ/N\ZZ)^*\).

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20,QQ)
sage: b.maximize_base_ring()
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> -1
sage: b.maximize_base_ring().base_ring()
Cyclotomic Field of order 4 and degree 2
sage: DirichletGroup(20).base_ring()
Cyclotomic Field of order 4 and degree 2
minimize_base_ring()

Return a Dirichlet character that equals this one, but over as small a subfield (or subring) of the base ring as possible.

Note

This function is currently only implemented when the base ring is a number field. It’s the identity function in characteristic p.

EXAMPLES:

sage: G = DirichletGroup(13)
sage: e = DirichletGroup(13).0
sage: e.base_ring()
Cyclotomic Field of order 12 and degree 4
sage: e.minimize_base_ring().base_ring()
Cyclotomic Field of order 12 and degree 4
sage: (e^2).minimize_base_ring().base_ring()
Cyclotomic Field of order 6 and degree 2
sage: (e^3).minimize_base_ring().base_ring()
Cyclotomic Field of order 4 and degree 2
sage: (e^12).minimize_base_ring().base_ring()
Rational Field
modulus()

The modulus of this character.

EXAMPLES:

sage: e = DirichletGroup(100, QQ).0
sage: e.modulus()
100
sage: e.conductor()
4
multiplicative_order()

The order of this character.

EXAMPLES:

sage: e = DirichletGroup(100).1
sage: e.order()    # same as multiplicative_order, since group is multiplicative
20
sage: e.multiplicative_order()
20
sage: e = DirichletGroup(100).0
sage: e.multiplicative_order()
2
primitive_character()

Returns the primitive character associated to self.

EXAMPLES:

sage: e = DirichletGroup(100).0; e
Dirichlet character modulo 100 of conductor 4 mapping 51 |--> -1, 77 |--> 1
sage: e.conductor()
4
sage: f = e.primitive_character(); f
Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1
sage: f.modulus()
4
restrict(M)

Returns the restriction of this character to a Dirichlet character modulo the divisor M of the modulus, which must also be a multiple of the conductor of this character.

EXAMPLES:

sage: e = DirichletGroup(100).0
sage: e.modulus()
100
sage: e.conductor()
4
sage: e.restrict(20)
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: e.restrict(4)
Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1
sage: e.restrict(50)
Traceback (most recent call last):
...
ValueError: conductor(=4) must divide M(=50)
values()

Return a list of the values of this character on each integer between 0 and the modulus.

EXAMPLES:

sage: e = DirichletGroup(20)(1)
sage: e.values()
[0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1]
sage: e = DirichletGroup(20).gen(0)
sage: e.values()
[0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1]
sage: e = DirichletGroup(20).gen(1)
sage: e.values()
[0, 1, 0, -zeta4, 0, 0, 0, zeta4, 0, -1, 0, 1, 0, -zeta4, 0, 0, 0, zeta4, 0, -1]
sage: e = DirichletGroup(21).gen(0) ; e.values()
[0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1]
sage: e = DirichletGroup(21, base_ring=GF(37)).gen(0) ; e.values()
[0, 1, 36, 0, 1, 36, 0, 0, 36, 0, 1, 36, 0, 1, 0, 0, 1, 36, 0, 1, 36]
sage: e = DirichletGroup(21, base_ring=GF(3)).gen(0) ; e.values()
[0, 1, 2, 0, 1, 2, 0, 0, 2, 0, 1, 2, 0, 1, 0, 0, 1, 2, 0, 1, 2]
sage: chi = DirichletGroup(100151, CyclotomicField(10)).0
sage: ls = chi.values() ; ls[0:10]
[0,
1,
-zeta10^3,
-zeta10,
-zeta10,
1,
zeta10^3 - zeta10^2 + zeta10 - 1,
zeta10,
zeta10^3 - zeta10^2 + zeta10 - 1,
zeta10^2]
values_on_gens()

Return a tuple of the values of self on the standard generators of \((\ZZ/N\ZZ)^*\), where \(N\) is the modulus.

EXAMPLES:

sage: e = DirichletGroup(16)([-1, 1])
sage: e.values_on_gens ()
(-1, 1)

Note

The constructor of DirichletCharacter sets the cache of element() or of values_on_gens(). The cache of one of these methods needs to be set for the other method to work properly, these caches have to be stored when pickling an instance of DirichletCharacter.

class sage.modular.dirichlet.DirichletGroupFactory

Bases: sage.structure.factory.UniqueFactory

Construct a group of Dirichlet characters modulo \(N\).

INPUT:

  • N – positive integer
  • base_ring – commutative ring; the value ring for the characters in this group (default: the cyclotomic field \(\QQ(\zeta_n)\), where \(n\) is the exponent of \((\ZZ/N\ZZ)^*\))
  • zeta – (optional) root of unity in base_ring
  • zeta_order – (optional) positive integer; this must be the order of zeta if both are specified
  • names – ignored (needed so G.<...> = DirichletGroup(...) notation works)
  • integral – boolean (default: False); whether to replace the default cyclotomic field by its rings of integers as the base ring. This is ignored if base_ring is not None.

OUTPUT:

The group of Dirichlet characters modulo \(N\) with values in a subgroup \(V\) of the multiplicative group \(R^*\) of base_ring. This is the group of homomorphisms \((\ZZ/N\ZZ)^* \to V\) with pointwise multiplication. The group \(V\) is determined as follows:

  • If both zeta and zeta_order are omitted, then \(V\) is taken to be \(R^*\), or equivalently its \(n\)-torsion subgroup, where \(n\) is the exponent of \((\ZZ/N\ZZ)^*\). Many operations, such as finding a set of generators for the group, are only implemented if \(V\) is cyclic and a generator for \(V\) can be found.
  • If zeta is specified, then \(V\) is taken to be the cyclic subgroup of \(R^*\) generated by zeta. If zeta_order is also given, it must be the multiplicative order of zeta; this is useful if the base ring is not exact or if the order of zeta is very large.
  • If zeta is not specified but zeta_order is, then \(V\) is taken to be the group of roots of unity of order dividing zeta_order in \(R\). In this case, \(R\) must be a domain (so \(V\) is cyclic), and \(V\) must have order zeta_order. Furthermore, a generator zeta of \(V\) is computed, and an error is raised if such zeta cannot be found.

EXAMPLES:

The default base ring is a cyclotomic field of order the exponent of \((\ZZ/N\ZZ)^*\):

sage: DirichletGroup(20)
Group of Dirichlet characters modulo 20 with values in Cyclotomic Field of order 4 and degree 2

We create the group of Dirichlet character mod 20 with values in the rational numbers:

sage: G = DirichletGroup(20, QQ); G
Group of Dirichlet characters modulo 20 with values in Rational Field
sage: G.order()
4
sage: G.base_ring()
Rational Field

The elements of G print as lists giving the values of the character on the generators of \((Z/NZ)^*\):

sage: list(G)
[Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1, Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1, Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> -1, Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> -1]

Next we construct the group of Dirichlet character mod 20, but with values in \(\QQ(\zeta_n)\):

sage: G = DirichletGroup(20)
sage: G.1
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4

We next compute several invariants of G:

sage: G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1, Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4)
sage: G.unit_gens()
(11, 17)
sage: G.zeta()
zeta4
sage: G.zeta_order()
4

In this example we create a Dirichlet group with values in a number field:

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^4 + 1)
sage: DirichletGroup(5, K)
Group of Dirichlet characters modulo 5 with values in Number Field in a with defining polynomial x^4 + 1

An example where we give zeta, but not its order:

sage: G = DirichletGroup(5, K, a); G
Group of Dirichlet characters modulo 5 with values in the group of order 8 generated by a in Number Field in a with defining polynomial x^4 + 1
sage: G.list()
[Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1, Dirichlet character modulo 5 of conductor 5 mapping 2 |--> a^2, Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1, Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -a^2]

We can also restrict the order of the characters, either with or without specifying a root of unity:

sage: DirichletGroup(5, K, zeta=-1, zeta_order=2)
Group of Dirichlet characters modulo 5 with values in the group of order 2 generated by -1 in Number Field in a with defining polynomial x^4 + 1
sage: DirichletGroup(5, K, zeta_order=2)
Group of Dirichlet characters modulo 5 with values in the group of order 2 generated by -1 in Number Field in a with defining polynomial x^4 + 1
sage: G.<e> = DirichletGroup(13)
sage: loads(G.dumps()) == G
True
sage: G = DirichletGroup(19, GF(5))
sage: loads(G.dumps()) == G
True

We compute a Dirichlet group over a large prime field:

sage: p = next_prime(10^40)
sage: g = DirichletGroup(19, GF(p)); g
Group of Dirichlet characters modulo 19 with values in Finite Field of size 10000000000000000000000000000000000000121

Note that the root of unity has small order, i.e., it is not the largest order root of unity in the field:

sage: g.zeta_order()
2
sage: r4 = CyclotomicField(4).ring_of_integers()
sage: G = DirichletGroup(60, r4)
sage: G.gens()
(Dirichlet character modulo 60 of conductor 4 mapping 31 |--> -1, 41 |--> 1, 37 |--> 1, Dirichlet character modulo 60 of conductor 3 mapping 31 |--> 1, 41 |--> -1, 37 |--> 1, Dirichlet character modulo 60 of conductor 5 mapping 31 |--> 1, 41 |--> 1, 37 |--> zeta4)
sage: val = G.gens()[2].values_on_gens()[2] ; val
zeta4
sage: parent(val)
Gaussian Integers in Cyclotomic Field of order 4 and degree 2
sage: r4.residue_field(r4.ideal(29).factor()[0][0])(val)
17
sage: r4.residue_field(r4.ideal(29).factor()[0][0])(val) * GF(29)(3)
22
sage: r4.residue_field(r4.ideal(29).factor()[0][0])(G.gens()[2].values_on_gens()[2]) * 3
22
sage: parent(r4.residue_field(r4.ideal(29).factor()[0][0])(G.gens()[2].values_on_gens()[2]) * 3)
Residue field of Fractional ideal (-2*zeta4 + 5)
sage: DirichletGroup(60, integral=True)
Group of Dirichlet characters modulo 60 with values in Gaussian Integers in Cyclotomic Field of order 4 and degree 2
sage: parent(DirichletGroup(60, integral=True).gens()[2].values_on_gens()[2])
Gaussian Integers in Cyclotomic Field of order 4 and degree 2

If the order of zeta cannot be determined automatically, we can specify it using zeta_order:

sage: DirichletGroup(7, CC, zeta=exp(2*pi*I/6))
Traceback (most recent call last):
...
NotImplementedError: order of element not known

sage: DirichletGroup(7, CC, zeta=exp(2*pi*I/6), zeta_order=6)
Group of Dirichlet characters modulo 7 with values in the group of order 6 generated by 0.500000000000000 + 0.866025403784439*I in Complex Field with 53 bits of precision

If the base ring is not a domain (in which case the group of roots of unity is not necessarily cyclic), some operations still work, such as creation of elements:

sage: G = DirichletGroup(5, Zmod(15)); G
Group of Dirichlet characters modulo 5 with values in Ring of integers modulo 15
sage: chi = G([13]); chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 13
sage: chi^2
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 4
sage: chi.multiplicative_order()
4

Other operations only work if zeta is specified:

sage: G.gens()
Traceback (most recent call last):
...
NotImplementedError: factorization of polynomials over rings with composite characteristic is not implemented
sage: G = DirichletGroup(5, Zmod(15), zeta=2); G
Group of Dirichlet characters modulo 5 with values in the group of order 4 generated by 2 in Ring of integers modulo 15
sage: G.gens()
(Dirichlet character modulo 5 of conductor 5 mapping 2 |--> 2,)
create_key(N, base_ring=None, zeta=None, zeta_order=None, names=None, integral=False)

Create a key that uniquely determines a Dirichlet group.

create_object(version, key, **extra_args)

Create the object from the key (extra arguments are ignored). This is only called if the object was not found in the cache.

class sage.modular.dirichlet.DirichletGroup_class(base_ring, modulus, zeta, zeta_order)

Bases: sage.misc.fast_methods.WithEqualityById, sage.structure.parent.Parent

Group of Dirichlet characters modulo \(N\) with values in a ring \(R\).

Element

alias of DirichletCharacter

base_extend(R)

Return the base extension of self to R.

INPUT:

  • R – either a ring admitting a coercion map from the base ring of self, or a ring homomorphism with the base ring of self as its domain

EXAMPLES:

sage: G = DirichletGroup(7,QQ); G
Group of Dirichlet characters modulo 7 with values in Rational Field
sage: H = G.base_extend(CyclotomicField(6)); H
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2

Note that the root of unity can change:

sage: H.zeta()
zeta6

This method (in contrast to change_ring()) requires a coercion map to exist:

sage: G.base_extend(ZZ)
Traceback (most recent call last):
...
TypeError: no coercion map from Rational Field to Integer Ring is defined

Base-extended Dirichlet groups do not silently get roots of unity with smaller order than expected (trac ticket #6018):

sage: G = DirichletGroup(10, QQ).base_extend(CyclotomicField(4))
sage: H = DirichletGroup(10, CyclotomicField(4))
sage: G is H
True

sage: G3 = DirichletGroup(31, CyclotomicField(3))
sage: G5 = DirichletGroup(31, CyclotomicField(5))
sage: K30 = CyclotomicField(30)
sage: G3.gen(0).base_extend(K30) * G5.gen(0).base_extend(K30)
Dirichlet character modulo 31 of conductor 31 mapping 3 |--> -zeta30^7 + zeta30^5 + zeta30^4 + zeta30^3 - zeta30 - 1

When a root of unity is specified, base extension still works if the new base ring is not an integral domain:

sage: f = DirichletGroup(17, ZZ, zeta=-1).0
sage: g = f.base_extend(Integers(15))
sage: g(3)
14
sage: g.parent().zeta()
14
change_ring(R, zeta=None, zeta_order=None)

Return the base extension of self to R.

INPUT:

  • R – either a ring admitting a conversion map from the base ring of self, or a ring homomorphism with the base ring of self as its domain
  • zeta – (optional) root of unity in R
  • zeta_order – (optional) order of zeta

EXAMPLES:

sage: G = DirichletGroup(7,QQ); G
Group of Dirichlet characters modulo 7 with values in Rational Field
sage: G.change_ring(CyclotomicField(6))
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2
decomposition()

Returns the Dirichlet groups of prime power modulus corresponding to primes dividing modulus.

(Note that if the modulus is 2 mod 4, there will be a “factor” of \((\ZZ/2\ZZ)^*\), which is the trivial group.)

EXAMPLES:

sage: DirichletGroup(20).decomposition()
[
Group of Dirichlet characters modulo 4 with values in Cyclotomic Field of order 4 and degree 2,
Group of Dirichlet characters modulo 5 with values in Cyclotomic Field of order 4 and degree 2
]
sage: DirichletGroup(20,GF(5)).decomposition()
[
Group of Dirichlet characters modulo 4 with values in Finite Field of size 5,
Group of Dirichlet characters modulo 5 with values in Finite Field of size 5
]
exponent()

Return the exponent of this group.

EXAMPLES:

sage: DirichletGroup(20).exponent()
4
sage: DirichletGroup(20,GF(3)).exponent()
2
sage: DirichletGroup(20,GF(2)).exponent()
1
sage: DirichletGroup(37).exponent()
36
galois_orbits(v=None, reps_only=False, sort=True, check=True)

Return a list of the Galois orbits of Dirichlet characters in self, or in v if v is not None.

INPUT:

  • v - (optional) list of elements of self
  • reps_only - (optional: default False) if True only returns representatives for the orbits.
  • sort - (optional: default True) whether to sort the list of orbits and the orbits themselves (slightly faster if False).
  • check - (optional, default: True) whether or not to explicitly coerce each element of v into self.

The Galois group is the absolute Galois group of the prime subfield of Frac(R). If R is not a domain, an error will be raised.

EXAMPLES:

sage: DirichletGroup(20).galois_orbits()
[
[Dirichlet character modulo 20 of conductor 20 mapping 11 |--> -1, 17 |--> -1],
...,
[Dirichlet character modulo 20 of conductor 1 mapping 11 |--> 1, 17 |--> 1]
]
sage: DirichletGroup(17, Integers(6), zeta=Integers(6)(5)).galois_orbits()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain
sage: DirichletGroup(17, Integers(9), zeta=Integers(9)(2)).galois_orbits()
Traceback (most recent call last):
...
TypeError: Galois orbits only defined if base ring is an integral domain
gen(n=0)

Return the n-th generator of self.

EXAMPLES:

sage: G = DirichletGroup(20)
sage: G.gen(0)
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: G.gen(1)
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4
sage: G.gen(2)
Traceback (most recent call last):
...
IndexError: n(=2) must be between 0 and 1
sage: G.gen(-1)
Traceback (most recent call last):
...
IndexError: n(=-1) must be between 0 and 1
gens()

Returns generators of self.

EXAMPLES:

sage: G = DirichletGroup(20)
sage: G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1, Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> zeta4)
integers_mod()

Returns the group of integers \(\ZZ/N\ZZ\) where \(N\) is the modulus of self.

EXAMPLES:

sage: G = DirichletGroup(20)
sage: G.integers_mod()
Ring of integers modulo 20
list()

Return a list of the Dirichlet characters in this group.

EXAMPLES:

sage: DirichletGroup(5).list()
[Dirichlet character modulo 5 of conductor 1 mapping 2 |--> 1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -1,
 Dirichlet character modulo 5 of conductor 5 mapping 2 |--> -zeta4]
modulus()

Returns the modulus of self.

EXAMPLES:

sage: G = DirichletGroup(20)
sage: G.modulus()
20
ngens()

Returns the number of generators of self.

EXAMPLES:

sage: G = DirichletGroup(20)
sage: G.ngens()
2
order()

Return the number of elements of self. This is the same as len(self).

EXAMPLES:

sage: DirichletGroup(20).order()
8
sage: DirichletGroup(37).order()
36
random_element()

Return a random element of self.

The element is computed by multiplying a random power of each generator together, where the power is between 0 and the order of the generator minus 1, inclusive.

EXAMPLES:

sage: DirichletGroup(37).random_element()
Dirichlet character modulo 37 of conductor 37 mapping 2 |--> zeta36^4
sage: DirichletGroup(20).random_element()
Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1
sage: DirichletGroup(60).random_element()
Dirichlet character modulo 60 of conductor 3 mapping 31 |--> 1, 41 |--> -1, 37 |--> 1
unit_gens()

Returns the minimal generators for the units of \((\ZZ/N\ZZ)^*\), where \(N\) is the modulus of self.

EXAMPLES:

sage: DirichletGroup(37).unit_gens()
(2,)
sage: DirichletGroup(20).unit_gens()
(11, 17)
sage: DirichletGroup(60).unit_gens()
(31, 41, 37)
sage: DirichletGroup(20,QQ).unit_gens()
(11, 17)
zeta()

Return the chosen root of unity in the base ring.

EXAMPLES:

sage: DirichletGroup(37).zeta()
zeta36
sage: DirichletGroup(20).zeta()
zeta4
sage: DirichletGroup(60).zeta()
zeta4
sage: DirichletGroup(60,QQ).zeta()
-1
sage: DirichletGroup(60, GF(25,'a')).zeta()
2
zeta_order()

Return the order of the chosen root of unity in the base ring.

EXAMPLES:

sage: DirichletGroup(20).zeta_order()
4
sage: DirichletGroup(60).zeta_order()
4
sage: DirichletGroup(60, GF(25,'a')).zeta_order()
4
sage: DirichletGroup(19).zeta_order()
18
sage.modular.dirichlet.TrivialCharacter(N, base_ring=Rational Field)

Return the trivial character of the given modulus, with values in the given base ring.

EXAMPLES:

sage: t = trivial_character(7)
sage: [t(x) for x in [0..20]]
[0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]
sage: t(1).parent()
Rational Field
sage: trivial_character(7, Integers(3))(1).parent()
Ring of integers modulo 3
sage.modular.dirichlet.is_DirichletCharacter(x)

Return True if x is of type DirichletCharacter.

EXAMPLES:

sage: from sage.modular.dirichlet import is_DirichletCharacter
sage: is_DirichletCharacter(trivial_character(3))
True
sage: is_DirichletCharacter([1])
False
sage.modular.dirichlet.is_DirichletGroup(x)

Returns True if x is a Dirichlet group.

EXAMPLES:

sage: from sage.modular.dirichlet import is_DirichletGroup
sage: is_DirichletGroup(DirichletGroup(11))
True
sage: is_DirichletGroup(11)
False
sage: is_DirichletGroup(DirichletGroup(11).0)
False
sage.modular.dirichlet.kronecker_character(d)

Return the quadratic Dirichlet character (d/.) of minimal conductor.

EXAMPLES:

sage: kronecker_character(97*389*997^2)
Dirichlet character modulo 37733 of conductor 37733 mapping 1557 |--> -1, 37346 |--> -1
sage: a = kronecker_character(1)
sage: b = DirichletGroup(2401,QQ)(a)    # NOTE -- over QQ!
sage: b.modulus()
2401

AUTHORS:

  • Jon Hanke (2006-08-06)
sage.modular.dirichlet.kronecker_character_upside_down(d)

Return the quadratic Dirichlet character (./d) of conductor d, for d0.

EXAMPLES:

sage: kronecker_character_upside_down(97*389*997^2)
Dirichlet character modulo 37506941597 of conductor 37733 mapping 13533432536 |--> -1, 22369178537 |--> -1, 14266017175 |--> 1

AUTHORS:

  • Jon Hanke (2006-08-06)
sage.modular.dirichlet.trivial_character(N, base_ring=Rational Field)

Return the trivial character of the given modulus, with values in the given base ring.

EXAMPLES:

sage: t = trivial_character(7)
sage: [t(x) for x in [0..20]]
[0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1]
sage: t(1).parent()
Rational Field
sage: trivial_character(7, Integers(3))(1).parent()
Ring of integers modulo 3