↑

Properties

Label 15867.322
Modulus $15867$
Conductor $15867$
Order $840$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15867, base_ring=CyclotomicField(840)) M = H._module chi = DirichletCharacter(H, M([560,441,720]))
 
Copy content gp:[g,chi] = znchar(Mod(322, 15867))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15867.322");
 

Basic properties

Modulus: \(15867\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(15867\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(840\)
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: yes
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 15867.mg

\(\chi_{15867}(97,\cdot)\) \(\chi_{15867}(193,\cdot)\) \(\chi_{15867}(274,\cdot)\) \(\chi_{15867}(322,\cdot)\) \(\chi_{15867}(391,\cdot)\) \(\chi_{15867}(403,\cdot)\) \(\chi_{15867}(520,\cdot)\) \(\chi_{15867}(580,\cdot)\) \(\chi_{15867}(637,\cdot)\) \(\chi_{15867}(643,\cdot)\) \(\chi_{15867}(709,\cdot)\) \(\chi_{15867}(772,\cdot)\) \(\chi_{15867}(790,\cdot)\) \(\chi_{15867}(895,\cdot)\) \(\chi_{15867}(967,\cdot)\) \(\chi_{15867}(1096,\cdot)\) \(\chi_{15867}(1129,\cdot)\) \(\chi_{15867}(1159,\cdot)\) \(\chi_{15867}(1165,\cdot)\) \(\chi_{15867}(1177,\cdot)\) \(\chi_{15867}(1258,\cdot)\) \(\chi_{15867}(1282,\cdot)\) \(\chi_{15867}(1411,\cdot)\) \(\chi_{15867}(1483,\cdot)\) \(\chi_{15867}(1546,\cdot)\)

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: $\Q(\zeta_{840})$
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 840 polynomial (not computed)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((14105,15094,7012)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{21}{40}\right),e\left(\frac{6}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 15867 }(322, a) \) \(-1\)\(1\)\(e\left(\frac{193}{420}\right)\)\(e\left(\frac{193}{210}\right)\)\(e\left(\frac{131}{420}\right)\)\(e\left(\frac{17}{120}\right)\)\(e\left(\frac{53}{140}\right)\)\(e\left(\frac{27}{35}\right)\)\(e\left(\frac{803}{840}\right)\)\(e\left(\frac{31}{840}\right)\)\(e\left(\frac{101}{168}\right)\)\(e\left(\frac{88}{105}\right)\)
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)
 
\( \chi_{ 15867 }(322,a) \;\) at \(\;a = \) e.g. 2