Let R be a ring. Then a Dirichlet character over R of modulus N is a homomorphism ε : (Z/NZ) * -> R * , where R * is the group of invertible elements of R. We extend ε to a set theoretic map on the whole of Z by defining ε(x) = 0 if gcd(x, N) != 1. The conductor of ε 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) * .
The group of Dirichlet characters modulo N with values in RationalField(). (Note that this is a group of exponent at most 2.)
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.
The group of Dirichlet characters modulo N with values in the ring R. Here R can be the integers, rationals, a number field or a finite field. (Note that this group may be smaller than the full Dirichlet group.)
The group of Dirichlet characters mod N with values in the cyclic subgroup generated by the root of unity z in the ring R. Here z must be an element of R of exact order r (where r may be smaller than the exponent of the full Dirichlet group).
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 is the fundamental discriminant associated to the integer D.When a ring R is given, this is returned as a character with values in 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.
The number of generators of the Dirichlet group G.
A sequence containing generators for the Dirichlet 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.
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.
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 ring in which the Dirichlet character χ takes values.
The modulus of the group of Dirichlet characters that contains χ.
The minimal conductor of the Dirichlet character χ. (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 χ (in terms of generators of the group containing χ).
Return true iff the given characters have the same modulus and values.
The order of the given element χ in a group of Dirichlet characters.
Returns true if and only if the Dirichlet character χ has order 1.
Returns true iff the Dirichlet character χ is primitive (equivalently, if its conductor equals its modulus).
The primitive character modulo the conductor of χ which takes the same values (on units) as χ.
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 χ is zero if χ 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 χ is zero if χ is odd and k is even.
For a Dirichlet character χ, this is true if and only if every character in the Decomposition of χ (into prime power components) is even.
This decomposes the Dirichlet character χ 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).
The returns a character which is the same as χ, except which takes values in the smallest possible subring of the base ring of χ.
The value of the Dirichlet character χ at the integer n.
A sequence containing the values [χ(1), .., χ(N)] of the given character χ, where N is the modulus of χ.The list of values is stored; then in later calls to Evaluate, the stored value is returned.
A sequence containing the values of χ on the ordered sequence of elements of Z/m given by UnitGenerators(Parent(chi)), where m is the modulus of χ.
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.)
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).
> 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); falseNext 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); trueCharacters 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_8^2If 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)