↑

Properties

Label 2156.bs
Modulus $2156$
Conductor $196$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2156, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([21,38,0])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(23, 2156)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2156.23"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(2156\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(196\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(42\)
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 196.o
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);
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.74252462132603256348231837398371002884673933378885582779211491265789772693504.1

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(9\) \(13\) \(15\) \(17\) \(19\) \(23\) \(25\) \(27\)
\(\chi_{2156}(23,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{2156}(331,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{2156}(375,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{2156}(639,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{2156}(683,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{2156}(947,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{2156}(991,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{2156}(1299,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{2156}(1563,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{2156}(1607,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{2156}(1871,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{2156}(1915,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{3}{14}\right)\)