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Let R be a ring. Then a Dirichlet character
over R of modulus N is a homomorphism
varepsilon : (Z/NZ)^ * -> R^ * ,
where R^ * is the group of invertible elements of R.
We extend varepsilon to a set theoretic map on the whole
of Z by defining varepsilon(x) = 0 if gcd(x, N) != 1.
The conductor of varepsilon is the smallest positive
integer M such that the homomorphism
(Z/NZ)^ * -> R^ *
factors through (Z/MZ)^ * via the natural
map (Z/NZ)^ * -> (Z/MZ)^ *.
Subsections
The group of Dirichlet characters modulo N with image in RationalField().
Note that this is a group of exponent at most 2.
The group of Dirichlet characters modulo N with image in the ring R.
Here R can be the integers, rationals, a number field or a finite field.
The group of Dirichlet characters mod N with image in the
order-r cyclic subgroup of the ring R generated by the root of unity z.
Here z must be an element of R of exact order r.
The group of Dirichlet characters modulo N taking values
in the mth cyclotomic field, where m is the exponent of the
unit group modulo N. (This is a shortcut for the previous command.)
BaseExtend(G, R, z) : GrpDrch, Rng, RngElt -> GrpDrch
The group of Dirichlet characters corresponding to G with values
in the ring R. In the second form, the distinguished
root of unity of the base ring of G is identified with the
given element z.
Assign names to the generators of the Dirichlet group G.
A sequence containing all Dirichlet characters in the Dirichlet group G.
A random element of the Dirichlet group G.
The ith generator of the group G.
This coerces the given element x into the Dirichlet group G.
Here x may be a Dirichlet character belonging to a different
group, or a sequence of integers specifying an element of the
AbelianGroup of G.
The Kronecker character n |-> (D/n), where D can be any integer.
Thus Evaluate(KroneckerCharacter(D), n) equals KroneckerSymbol(D,n).
The Kronecker character n |-> (D/n) over the ring R.
The ring in which characters in G take values.
The integer N such that G is a group of Dirichlet characters
on Z/N.
The order of the Dirichlet group G.
The exponent of the Dirichlet group G.
This returns a finite abelian group isomorphic to the given group G of
Dirichlet characters (as an abstract group), and secondly returns a map
from the abstract group to G.
It is necessary to use this function in order to make group theoretic
constructions involving G.
The number of generators of the Dirichlet group G.
A sequence containing generators for the Dirichlet group G.
The ith generator of the group G.
This returns an ordered sequence of integers that reduce to
"canonical" generators of the unit group of Z/N,
where N is the modulus of G.
The ring in which the Dirichlet character chi takes values.
The modulus of the group of Dirichlet characters that contains chi.
The minimal conductor of the Dirichlet character chi.
(That is, the smallest integer M such that chi is
well-defined on the unit group of Z/M.)
A sequence of integers specifying the Dirichlet character chi
(in terms of generators of the group containing chi).
Return true iff the given characters have the same modulus and values.
The order of the given element chi in a group of Dirichlet characters.
Returns true if and only if the Dirichlet character chi has order 1.
Returns true iff the Dirichlet character chi is primitive
(equivalently, if its conductor equals its modulus).
The primitive character modulo the conductor of chi which takes
the same values (on units) as chi.
Returns true if and only if Evaluate(chi,-1) is equal to 1.
Note that in characteristic 0,
the space of modular forms of weight k and character chi
is zero if chi is even and k is odd.
Returns true if and only if Evaluate(chi,-1) is equal to -1.
Note that in characteristic 0, the space of modular
forms of weight k and character chi is zero if chi
is odd and k is even.
For a Dirichlet character chi, this is true if and only if
every character in the Decomposition of chi (into prime power
components) is even.
This decomposes the Dirichlet character chi as a product
of characters with prime power moduli. The function returns
a list (not a sequence) containing these characters (which
do not belong to the same group).
GaloisConjugacyRepresentatives(seq) : [GrpDrchElt] -> [GrpDrchElt]
This returns a sequence containing one representative from each
Galois conjugacy class (over Q) of characters corresponding
to a character in the given group or the given sequence.
The returns a character which is the same as chi, except which takes
values in the smallest possible subring of the base ring of chi.
chi(n) : GrpDrchElt, RngIntElt -> RngElt
The value of the Dirichlet character chi at the integer n.
A sequence containing the values [chi(1), .., chi(N)]
of the given character chi, where N is the modulus of chi.
The list of values is stored; then in later calls to Evaluate,
the stored value is returned.
A sequence containing the values of chi on the ordered
sequence of elements of Z/m given by UnitGenerators(Parent(chi)),
where m is the modulus of chi.
Given an element r of some ring which is assumed to satisfy rn = 1,
this returns the smallest integer m such that rm = 1.
(This provides a convenient way to calculate the order of values of
non-real characters.)
x / y : GrpDrchElt, GrpDrchElt -> GrpDrchElt
The product or quotient (respectively) of the Dirichlet characters x and y.
This is a Dirichlet character of modulus equal to the least common multiple
of the moduli of x and y. The base rings and chosen roots of unity of
the parents of x and y are equal.
The Dirichlet character x raised to the power of n, where n is any integer.
The image of the Dirichlet character x under the automorphism φ.
Given a Dirichlet character x of odd order, this returns a square root
of x (in the same group).
We begin by constructing the group of characters
(Z/5Z)^ * -> Q^ *.
> G<a> := DirichletGroup(5); G; // The default base field is Q.
Group of Dirichlet characters of modulus 5 over Rational Field
> #G;
2
> [Evaluate(a, n) : n in [1..5]];
[ 1, -1, -1, 1, 0 ]
> Eltseq(a);
[ 2 ]
> a eq G![2];
true
> IsEven(a);
true
> IsOdd(a);
false
> IsTrivial(a);
false
Next we create a character by building it up "locally".
> G1<a4> := DirichletGroup(4);
> Conductor(a4);
4
> G2<a5> := DirichletGroup(25);
> Conductor(a5);
5
> eps := a4*a5;
> Modulus(eps);
100
> Conductor(eps);
20
> Evaluate(eps,7) eq Evaluate(a4,7)*Evaluate(a5,7);
true
Characters can be constructed over various fields.
> G<a> := DirichletGroup(7,GF(7));
> #G;
6
> Evaluate(a,2);
2
>
> G<a3,a5> := DirichletGroup(15,CyclotomicField(EulerPhi(15)));
> G;
Group of Dirichlet characters of modulus 15 over Cyclotomic Field of
order 8 and degree 4
> #G;
8
> Conductor(a3);
3
> Conductor(a5);
5
> Order(a5);
4
> Evaluate(a5,2);
zeta_8^2
If D is a fundamental discriminant, then KroneckerCharacter(D)
is the quadratic Dirichlet character corresponding to the quadratic
field Q(Sqrt(D)). The following code verifies that
KroneckerCharacter and KroneckerSymbol agree
in the case D=209.
> chi := KroneckerCharacter(209);
> for n in [1..209] do
> assert Evaluate(chi,n) eq KroneckerSymbol(209,n);
> end for;
If E is an elliptic curve with newform fE, then the twist ED
corresponds to fE twisted by this character, as illustrated below.
> E := EllipticCurve(CremonaDatabase(),"11A");
> f := qEigenform(E,8); f;
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
> chi := KroneckerCharacter(-7);
> qEigenform(QuadraticTwist(E,-7),8);
q - 2*q^2 + q^3 + 2*q^4 - q^5 - 2*q^6 + O(q^8)
> R<q> := Parent(f);
> &+[Evaluate(chi,n)*Coefficient(f,n)*q^n : n in [1..7]] + O(q^8);
q - 2*q^2 + q^3 + 2*q^4 - q^5 - 2*q^6 + O(q^8)
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