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
[zeta12^3, 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
[-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
class sage.modular.dirichlet.DirichletCharacter(parent, x, check=True)

A Dirichlet character

_DirichletCharacter__eval_at_minus_one()

Efficiently evaluate the character at -1 using knowledge of its order. This is potentially much more efficient than computing the value of -1 directly using dlog and a large power of the image root of unity.

We use the following. Proposition: Suppose eps is a character mod p^n, where p is a prime. Then \varepsilon(-1) = -1 if and only if p = 2 and the factor of eps at 4 is nontrivial or p > 2 and 2 does not divide \phi(p^n)/\mbox{\rm ord}(\varepsilon).

EXAMPLES:

sage: chi = DirichletGroup(20).0; chi._DirichletCharacter__eval_at_minus_one()
-1
__call__(m)

Return the value of this character at the integer m.

Warning

A table of values of the character is made the first time you call this. This table is currently constructed in a somewhat stupid way, though it is still pretty fast.

EXAMPLES:

sage: G = DirichletGroup(60)
sage: e = prod(G.gens(), G(1))
sage: e
[-1, -1, zeta4]
sage: e(2)
0
sage: e(7)
-zeta4
sage: Integers(60).unit_gens()
[31, 41, 37]
sage: e(31)
-1
sage: e(41)
-1
sage: e(37)
zeta4
sage: e(31*37)
-zeta4
sage: parent(e(31*37))
Cyclotomic Field of order 4 and degree 2
__cmp__(other)

Compare self to other. Note that this only gets called when the parents of self and other are identical, via a canonical coercion map; this means that characters of different moduli compare as unequal, even if they define identical functions on ZZ.

EXAMPLES:

sage: e = DirichletGroup(16)([-1, 1])
sage: f = e.restrict(8)
sage: e == e
True
sage: f == f
True
sage: e == f
False
sage: k = DirichletGroup(7)([-1])
sage: k == e
False
__copy__()

Return a (shallow) copy of this Dirichlet character.

EXAMPLE:

sage: G.<a> = DirichletGroup(11)
sage: b = copy(a)
sage: a is b
False
sage: a.element() is b.element()
True
__hash__()

EXAMPLES:

sage: e = DirichletGroup(16)([-1, 1])
sage: hash(e)
1498523633                  # 32-bit
3713082714464823281         # 64-bit
__init__(parent, x, check=True)

Create with DirichletCharacter(parent, values_on_gens)

INPUT:

  • parent - DirichletGroup, a group of Dirichlet characters
  • x
    • tuple (or list) of ring elements, the values of the Dirichlet character on the chosen generators of (\ZZ/N\ZZ)^*.
    • Vector over Z/eZ, where e is the order of the root of unity.

OUTPUT:

  • DirichletCharacter - a Dirichlet character

EXAMPLES:

sage: G, e = DirichletGroup(13).objgen()
sage: G
Group of Dirichlet characters of modulus 13 over Cyclotomic Field of order 12 and degree 4
sage: e
[zeta12]
sage: loads(e.dumps()) == e
True
sage: G, x = DirichletGroup(35).objgens()
sage: e = x[0]*x[1]; e
[zeta12^3, zeta12^2]
sage: e.order()
12
sage: loads(e.dumps()) == e
True
__invert__()

Return the multiplicative inverse of self. The notation is self.

EXAMPLES:

sage: e = DirichletGroup(13).0
sage: f = ~e
sage: f*e
[1]
__pow__(n)

Return self raised to the power of n

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: a^2
[1, 1]
sage: b^2
[1, -1]
__weakref__
list of weak references to the object (if defined)
_mul_(other)

Return the product of self and other.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: a
[-1, 1]
sage: b
[1, zeta4]
sage: a*b # indirect doctest
[-1, zeta4]

Multiplying elements whose parents have different zeta orders works:

sage: a = DirichletGroup(3, QQ, zeta=1, zeta_order=1)(1)
sage: b = DirichletGroup(3, QQ, zeta=-1, zeta_order=2)([-1])
sage: a * b # indirect doctest
[-1]
_repr_()

String representation of self.

EXAMPLES:

sage: G.<a,b> = DirichletGroup(20)
sage: repr(a) # indirect doctest
'[-1, 1]'
bar()

Return the complex conjugate of this Dirichlet character.

EXAMPLES:

sage: e = DirichletGroup(5).0
sage: e
[zeta4]
sage: e.bar()
[-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 - an integer
  • algorithm - string (default: ‘recurrence’); either ‘recurrence’ or ‘definition’. The ‘recurrence’ algorithm expresses generalized Bernoulli numbers in terms of classical Bernoulli numbers using a recurrence formula and is usually optimal. In this case **opts is passed onto the bernoulli function.
  • cache - if True, cache answers

Let eps be this character (not necessarily primitive), and let k \geq 0 be an integer weight. This function computes the (generalized) Bernoulli number B_{k,eps}, e.g., as defined on page 44 of Diamond-Im:

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

where N is the modulus of \varepsilon.

The default algorithm is the recurrence on page 656 of Cohen’s GTM ‘Number Theory and Diophantine Equations’, section 9.

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)

Returns the base extension of self to the ring R.

EXAMPLE:

sage: e = DirichletGroup(7, QQ).0
sage: f = e.change_ring(QuadraticField(3, 'a'))
sage: f.parent()
Group of Dirichlet characters of modulus 7 over Number Field in a with defining polynomial x^2 - 3
sage: e = DirichletGroup(13).0
sage: e.change_ring(QQ)
...
ValueError: cannot coerce element of order 12 into self
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
[[-1], [zeta4]]
sage: d[0].parent()
Group of Dirichlet characters of modulus 4 over Cyclotomic Field of order 4 and degree 2
sage: d[1].parent()
Group of Dirichlet characters of modulus 5 over 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
...
TypeError: unsupported operand parent(s) for '*': 'Group of Dirichlet characters of modulus 4 over Cyclotomic Field of order 4 and degree 2' and 'Group of Dirichlet characters of modulus 5 over 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()
[[], [zeta6]]
sage: (DirichletGroup(36).0).decomposition()
[[-1], [1]]
sage: (DirichletGroup(72).0).decomposition()
[[-1, 1], [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 implemented yet.

EXAMPLE:

sage: G.<a,b> = DirichletGroup(20)
sage: a.element()
(2, 0)
sage: b.element()
(0, 1)
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)
[-1, 1]
sage: a
[-1, 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()
[[1, zeta4], [1, -zeta4]]

Another example:

sage: G = DirichletGroup(13)
sage: G.galois_orbits()
[
[[1]],
[[zeta12], [zeta12^3 - zeta12], [-zeta12], [-zeta12^3 + zeta12]],
[[zeta12^2], [-zeta12^2 + 1]],
[[zeta12^3], [-zeta12^3]],
[[zeta12^2 - 1], [-zeta12^2]],
[[-1]]
]
sage: e = G.0
sage: e
[zeta12]
sage: e.galois_orbit()
[[zeta12], [zeta12^3 - zeta12], [-zeta12], [-zeta12^3 + zeta12]]
sage: e = G.0^2; e
[zeta12^2]
sage: e.galois_orbit()
[[zeta12^2], [-zeta12^2 + 1]]

A non-example:

sage: chi = DirichletGroup(7, Integers(9), zeta = Integers(9)(2)).0
sage: chi.galois_orbit()
...
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, i.e., \zeta is self.parent().zeta().

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
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, i.e., \zeta is self.parent().zeta().

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: e = G.0
sage: e.gauss_sum_numerical()
-3.07497205... + 1.8826966926...*I
sage: abs(e.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]

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]

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
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)
...
NotImplementedError: Characters must be from the same Dirichlet Group.
sage: all_jacobi_sums = [(DP[i],DP[j],DP[i].jacobi_sum(DP[j]))             ...                       for i in range(p-1) for j in range(p-1)[i:]]
...
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 #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

TESTS:

This shows that ticket #6393 has been fixed:

sage: G = DirichletGroup(5); X = G.list(); Y = X[0]; Z = X[1]
sage: # Y is trivial and Z is quartic
sage: sum([Y(x)*Z(1-x) for x in IntegerModRing(5)])
-1
sage: # The value -1 above is the correct value of the Jacobi sum J(Y, Z).
sage: Y.jacobi_sum(Z); Z.jacobi_sum(Y)
-1
-1
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, Salie 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 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()
1.73205080757
sage: G = DirichletGroup(20) 
sage: e = G.1  
sage: e.kloosterman_sum_numerical(53,3,11) 
3.80422606518 - 3.80422606518*I            
level()

Synonym for modulus.

EXAMPLE:

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()
[1, -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
[-1, 1]
sage: e.conductor()
4
sage: f = e.primitive_character(); f
[-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)
[-1, 1]
sage: e.restrict(4)
[-1]
sage: e.restrict(50)
...
ValueError: conductor(=4) must divide M(=50)
values()

Returns 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: print 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()

Returns a tuple of the values of this character on each of the minimal generators of (\ZZ/N\ZZ)^*, where N is the modulus.

EXAMPLES:

sage: e = DirichletGroup(16)([-1, 1])
sage: e.values_on_gens ()
(-1, 1)
sage.modular.dirichlet.DirichletGroup(modulus, base_ring=None, zeta=None, zeta_order=None, names=None, integral=False)

The group of Dirichlet characters modulo N with values in the subgroup \langle \zeta_n\rangle of the multiplicative group of the base_ring. If the base_ring is omitted then we use \QQ(\zeta_n), where n is the exponent of (\ZZ/N\ZZ)^*. If \zeta is omitted then we compute and use a maximal-order zeta in base_ring, if possible.

INPUT:

  • modulus - int
  • base_ring - Ring (optional), where characters take their values (should be an integral domain).
  • zeta - Element (optional), element of base_ring; zeta is a root of unity
  • zeta_order - int (optional), the order of zeta
  • names - ignored (needed so G.... = DirichletGroup(...) notation works)
  • integral - boolean (default: False). If True, return the group with base_ring the ring of integers in the smallest choice of CyclotomicField. Ignored if base_ring is not None.

OUTPUT:

  • DirichletGroup - a group of Dirichlet characters.

EXAMPLES:

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

sage: DirichletGroup(20)
Group of Dirichlet characters of modulus 20 over 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 of modulus 20 over 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)
[[1, 1], [-1, 1], [1, -1], [-1, -1]]

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

sage: G = DirichletGroup(20)
sage: G.list()
[[1, 1], [-1, 1], [1, zeta4], [-1, zeta4], [1, -1], [-1, -1], [1, -zeta4], [-1, -zeta4]]

We next compute several invariants of G:

sage: G.gens()
([-1, 1], [1, zeta4])
sage: G.unit_gens()
[11, 17]
sage: G.zeta()
zeta4
sage: G.zeta_order()
4

In this example we create a Dirichlet character with values in a number field. We have to give zeta, but not its order.

sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^4 + 1)
sage: G = DirichletGroup(5, K, a); G
Group of Dirichlet characters of modulus 5 over Number Field in a with defining polynomial x^4 + 1
sage: G.list()
[[1], [a^2], [-1], [-a^2]]
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 of modulus 19 over 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()
([-1, 1, 1], [1, -1, 1], [1, 1, zeta4])
sage: val = G.gens()[2].values_on_gens()[2] ; val
zeta4
sage: parent(val)
Maximal Order 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 of modulus 60 over Maximal Order in Cyclotomic Field of order 4 and degree 2
sage: parent(DirichletGroup(60, integral=True).gens()[2].values_on_gens()[2])
Maximal Order in Cyclotomic Field of order 4 and degree 2
class sage.modular.dirichlet.DirichletGroup_class(modulus, zeta, zeta_order)

Group of Dirichlet characters modulo N over a given base ring R.

__call__(x)

Coerce x into this Dirichlet group.

EXAMPLES:

sage: G = DirichletGroup(13)
sage: K = G.base_ring()
sage: G(1)
[1]
sage: G([-1])
[-1]
sage: G([K.0])
[zeta12]
sage: G(0)
...
TypeError: No coercion of 0 into Group of Dirichlet characters of modulus 13 over Cyclotomic Field of order 12 and degree 4 defined.
__cmp__(other)

Compare two Dirichlet groups. They are equal if they have the same modulus, are over the same base ring, and have the same chosen root of unity. Otherwise we compare first on the modulus, then the base ring, and finally the root of unity.

EXAMPLES:

sage: DirichletGroup(13) == DirichletGroup(13)
True
sage: DirichletGroup(13) == DirichletGroup(13,QQ)
False
sage: DirichletGroup(11) < DirichletGroup(13,QQ)
True
sage: DirichletGroup(17) > DirichletGroup(13)
True
__init__(modulus, zeta, zeta_order)

Create a Dirichlet group. Not to be called directly (use the factory function DirichletGroup.)

EXAMPLES:

sage: G = DirichletGroup(7, base_ring = Integers(9), zeta = Integers(9)(2)) # indirect doctest
sage: G.base() # check that ParentWithBase.__init__ has been called
Ring of integers modulo 9
__len__()

Return the number of elements of this Dirichlet group. This is the same as self.order().

EXAMPLES:

sage: len(DirichletGroup(20))
8
sage: len(DirichletGroup(20, QQ))
4
sage: len(DirichletGroup(20, GF(5)))
8
sage: len(DirichletGroup(20, GF(2)))
1
sage: len(DirichletGroup(20, GF(3)))
4
_automorphisms()

Compute the automorphisms of self. These are always given by raising to a power, so the return value is a list of integers.

At present this is only implemented if the base ring has characteristic 0 or a prime.

EXAMPLES:

sage: DirichletGroup(17)._automorphisms()
[1, 3, 5, 7, 9, 11, 13, 15]
sage: DirichletGroup(17, GF(11^4, 'a'))._automorphisms()
[1, 11, 121, 1331]
sage: DirichletGroup(17, Integers(6), zeta=Integers(6)(5))._automorphisms()
...
NotImplementedError: Automorphisms for finite non-field base rings not implemented
sage: DirichletGroup(17, Integers(9), zeta=Integers(9)(2))._automorphisms()
...
NotImplementedError: Automorphisms for finite non-field base rings not implemented
_coerce_impl(x)

Canonical coercion of x into self.

Note that although there is conversion between Dirichlet groups of different moduli, there is never canonical coercion. The main result of this is that Dirichlet characters of different moduli never compare as equal.

TESTS:

sage: trivial_character(6) == trivial_character(3) # indirect doctest
False
sage: trivial_character(3) == trivial_character(9)
False
sage: trivial_character(3) == DirichletGroup(3, QQ).0^2
True
_coerce_in_dirichlet_character(x)

Create an element of this group from a Dirichlet character, whose conductor must divide the modulus of self.

EXAMPLES:

sage: G = DirichletGroup(6)
sage: G._coerce_in_dirichlet_character(DirichletGroup(3).0)
[-1]
sage: G._coerce_in_dirichlet_character(DirichletGroup(15).0)
[-1]
sage: G._coerce_in_dirichlet_character(DirichletGroup(15).1)
...
TypeError: conductor must divide modulus
sage: H = DirichletGroup(16, QQ); H._coerce_in_dirichlet_character(DirichletGroup(16).1)
...
ValueError: cannot coerce element of order 4 into self
_repr_()

Return a print representation of this group, which can be renamed.

EXAMPLES:

sage: G = DirichletGroup(11)
sage: repr(G) # indirect doctest
'Group of Dirichlet characters of modulus 11 over Cyclotomic Field of order 10 and degree 4'
sage: G.rename('Dir(11)')
sage: G
Dir(11)
base_extend(R)

Returns the Dirichlet group over R obtained by extending scalars, with the same modulus and root of unity as self.

EXAMPLES:

sage: G = DirichletGroup(7,QQ); G
Group of Dirichlet characters of modulus 7 over Rational Field
sage: H = G.base_extend(CyclotomicField(6)); H
Group of Dirichlet characters of modulus 7 over Cyclotomic Field of order 6 and degree 2
sage: H.zeta()
-1
sage: G.base_extend(ZZ)
...
TypeError: No coercion map from 'Rational Field' to 'Integer Ring' is defined.
change_ring(R, zeta=None, zeta_order=None)

Returns the Dirichlet group over R with the same modulus as self.

EXAMPLES:

sage: G = DirichletGroup(7,QQ); G
Group of Dirichlet characters of modulus 7 over Rational Field
sage: G.change_ring(CyclotomicField(6))
Group of Dirichlet characters of modulus 7 over 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 of modulus 4 over Cyclotomic Field of order 4 and degree 2,
Group of Dirichlet characters of modulus 5 over Cyclotomic Field of order 4 and degree 2
]        
sage: DirichletGroup(20,GF(5)).decomposition()
[
Group of Dirichlet characters of modulus 4 over Finite Field of size 5,
Group of Dirichlet characters of modulus 5 over 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()
[
[[1, 1]],
[[1, zeta4], [1, -zeta4]],
[[1, -1]],
[[-1, 1]],
[[-1, zeta4], [-1, -zeta4]],
[[-1, -1]]
]
sage: DirichletGroup(17, Integers(6), zeta=Integers(6)(5)).galois_orbits()
...
TypeError: Galois orbits only defined if base ring is an integral domain
sage: DirichletGroup(17, Integers(9), zeta=Integers(9)(2)).galois_orbits()
...
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)
[-1, 1]
sage: G.gen(1)
[1, zeta4]
sage: G.gen(2)
...
IndexError: n(=2) must be between 0 and 1
sage: G.gen(-1)
...
IndexError: n(=-1) must be between 0 and 1
gens()

Returns generators of self.

EXAMPLES:

sage: G = DirichletGroup(20)
sage: G.gens()
([-1, 1], [1, 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
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()
[zeta36^4]
sage: DirichletGroup(20).random_element()
[-1, 1]
sage: DirichletGroup(60).random_element()
[1, -1, 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()

Returns the chosen root zeta of unity in the base ring R.

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()

Returns the order of the chosen root zeta of unity in the base ring R.

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.

EXAMPLE:

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)

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

EXAMPLES:

sage: kronecker_character(97*389*997^2)
[-1, -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)

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

EXAMPLES:

sage: kronecker_character_upside_down(97*389*997^2)
[-1, -1, 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.

EXAMPLE:

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

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