7.2.1.1 DirichletCharacter Objects

class DirichletCharacter
The class of Dirichlet characters.
DirichletCharacter( parent, values_on_gens)

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 $(\Z/N\Z)^*$.
OUTPUT:
    DirichletCharacter -- a Dirichlet character

Instances of class DirichletCharacter have the following methods (in addition to inherited methods and special methods):

base_ring,$  $ bernoulli,$  $ change_base,$  $ change_base_ring,$  $ conductor,$  $ decomposition,$  $ extend,$  $ is_even,$  $ is_odd,$  $ is_trivial,$  $ maximize_base_ring,$  $ minimize_base_ring,$  $ modulus,$  $ order,$  $ parent,$  $ restrict,$  $ values,$  $ values_on_gens

These methods are defined as follows:

base_ring( )

Returns the base ring of this Dirichlet character.

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)

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

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:

$\displaystyle \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 $ .

change_base( R)

Returns the base extension of self to the ring R.

change_base_ring( R)

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

conductor( )

Computes and returns the conductor of this character.

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.

extend( M)

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

is_even( )

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

is_odd( )

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

is_trivial( )

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

maximize_base_ring( )

Let

$\displaystyle \varepsilon : (\mathbf{Z}/N\mathbf{Z})^* \to \mathbf{Q}(\zeta_n)
$

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

$\displaystyle \chi : (\mathbf{Z}/N\mathbf{Z})^* \to \mathbf{Q}(\zeta_m)
$

where $ m$ is the least common multiple of $ n$ and the exponent of $ (\mathbf{Z}/N\mathbf{Z})^*$ .

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.

modulus( )

The modulus of this character.

order( )

The order of this character.

parent( )

The parent of this character.

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.

values( )

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

values_on_gens( )

Returns a list of the values of this character on each of the minimal generators of $ (\mathbf{Z}/N\mathbf{Z})^*$ , where $ N$ is the modulus.

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