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
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:
) |
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
k) |
Returns the generalized Bernoulli number .
Let eps be this character (not necessarily primitive), and
let be an integer weight. This function computes
the (generalized) Bernoulli number
, e.g., as defined
on page 44 of Diamond-Im:
R) |
Returns the base extension of self to the ring R.
R) |
Tries to compute the Dirichlet character over R that has the same values as this character, and if successful returns that character.
) |
Computes and returns the conductor of this character.
) |
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.
M) |
Returns the extension of this character to a Dirichlet character modulo the multiple M of the modulus.
) |
Return True
if and only if
.
) |
Return True
if and only if
.
) |
Returns True
if this is the trivial character, i.e., has
order 1.
) |
Let
) |
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.
) |
The modulus of this character.
) |
The order of this character.
) |
The parent of this character.
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.
) |
Returns a list of the values of this character on each integer between 0 and the modulus.
) |
Returns a list of the values of this character on each of the
minimal generators of
, where
is the modulus.
Instances of class DirichletCharacter also have the following special methods:
m) |
other) |
) |
other) |
exp) |
left) |
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