Package sage :: Package modular :: Module dirichlet :: Class DirichletCharacter
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Class DirichletCharacter


A DirichletCharacter is the extension of a homomorphism
(Z/NZ)^* --> R^*, for some ring R, to a map Z/NZ --> R,
got by sending those x with (N,x)>1 to 0.

Create with
    DirichletCharacter(parent, values_on_gens)
INPUT:
    parent -- DirichletGroup, a group of Dirichlet characters
    values_on_gens -- list of ring elements, the values of the
                      Dirichlet character on the chosen generators
                      of (Z/NZ)^*.
OUTPUT:
    DirichletCharacter -- a Dirichlet character

Method Summary
  __init__(self, parent, values_on_gens)
  __call__(self, m)
  __cmp__(self, other)
  __hash__(self)
  __invert__(self)
  __mul__(self, other)
  __pow__(self, exp)
  __repr__(self)
  __rmul__(self, left)
  base_ring(self)
Returns the base ring of the parent of self.
  bernoulli(self, k)
Returns the generalized Bernoulli number B_{k,eps}.
  change_base(self, R)
Returns the base extension of self to the ring R.
  change_base_ring(self, R)
Tries to compute the Dirichlet character over R that has the same values as self, and if successful returns that character.
  conductor(self)
Computes and returns the conductor of self.
  decomposition(self)
Return the decomposition of self as a product of Dirichlet characters of prime power modulus, where the prime powers exactly divide the modulus of self.
  extend(self, M)
Returns the extension of self to a Dirichlet character modulo the multiple M of the modulus.
  is_even(self)
Return True if and only if self(-1) == 1.
  is_odd(self)
Return True if and only if self(-1) != 1.
  is_trivial(self)
Returns true if self is the trivial character, i.e., has order 1.
  maximize_base_ring(self)
Let eps : (Z/N)^* ----> Q(zeta_n) be a Dirichlet character.
  minimize_base_ring(self)
Return a Dirichlet character that equals this one, but over as small a subfield (or subring) of the base ring as possible.
  modulus(self)
Returns the modulus of self.
  order(self)
Returns the order of self.
  parent(self)
Returns the parent of self.
  restrict(self, M)
Returns the restriction of self to a Dirichlet character modulo the divisor M of the modulus, which must also be a multiple of the conductor of self.
  values(self)
Returns a list of the values of self on each integer between 0 and the modulus of self.
  values_on_gens(self)
Returns a list of the values of self on each of the minimal generators of (Z/NZ)^*, where N is the modulus of self.

Method Details

base_ring(self)

Returns the base ring of the parent of self.

bernoulli(self, k)

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

Let eps be this character (not necessarily primitive), and
let k>=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} eps(a) t*e^(at)/(e^(N*t)-1) 
         = sum_{k=0}^{\infty} B_{k,eps}/{k!}*t^k.
         
where N is the modulus of eps.

change_base(self, R)

Returns the base extension of self to the ring R.

change_base_ring(self, R)

Tries to compute the Dirichlet character over R that has the same values as self, and if successful returns that character.

conductor(self)

Computes and returns the conductor of self.

decomposition(self)

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

extend(self, M)

Returns the extension of self to a Dirichlet character modulo the multiple M of the modulus. which must also be a

is_even(self)

Return True if and only if self(-1) == 1.

is_odd(self)

Return True if and only if self(-1) != 1.

is_trivial(self)

Returns true if self is the trivial character, i.e., has order 1.

maximize_base_ring(self)

Let

       eps : (Z/N)^* ----> Q(zeta_n)
       
be a Dirichlet character.  This function returns an equal
Dirichlet character

       chi : (Z/N)^* ----> Q(zeta_m)
       
where m is LCM(n, exponent of (Z/N)^*).

minimize_base_ring(self)

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.

modulus(self)

Returns the modulus of self.

order(self)

Returns the order of self.

parent(self)

Returns the parent of self.

restrict(self, M)

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

values(self)

Returns a list of the values of self on each integer between 0 and the modulus of self.

values_on_gens(self)

Returns a list of the values of self on each of the minimal generators of (Z/NZ)^*, where N is the modulus of self.

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