Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1015, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([0,14,6]))
 
Copy content gp:[g,chi] = znchar(Mod(16, 1015))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1015.16");
 

Basic properties

Modulus: \(1015\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(203\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(21\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from \(\chi_{203}(16,\cdot)\)
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1015.bw

\(\chi_{1015}(16,\cdot)\) \(\chi_{1015}(81,\cdot)\) \(\chi_{1015}(226,\cdot)\) \(\chi_{1015}(256,\cdot)\) \(\chi_{1015}(326,\cdot)\) \(\chi_{1015}(401,\cdot)\) \(\chi_{1015}(431,\cdot)\) \(\chi_{1015}(471,\cdot)\) \(\chi_{1015}(571,\cdot)\) \(\chi_{1015}(576,\cdot)\) \(\chi_{1015}(716,\cdot)\) \(\chi_{1015}(886,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 21 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((407,871,176)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{1}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1015 }(16, a) \) \(1\)\(1\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{21}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 1015 }(16,a) \;\) at \(\;a = \) e.g. 2