Properties

Label 119.m
Modulus 119119
Conductor 119119
Order 1212
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(119, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,9]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(38,119))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: 119119
Conductor: 119119
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 1212
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: Q(ζ12)\Q(\zeta_{12})
Fixed field: 12.0.33498139941871322753.1

Characters in Galois orbit

Character 1-1 11 22 33 44 55 66 88 99 1010 1111 1212
χ119(38,)\chi_{119}(38,\cdot) 1-1 11 e(56)e\left(\frac{5}{6}\right) e(1112)e\left(\frac{11}{12}\right) e(23)e\left(\frac{2}{3}\right) e(712)e\left(\frac{7}{12}\right) i-i 1-1 e(56)e\left(\frac{5}{6}\right) e(512)e\left(\frac{5}{12}\right) e(1112)e\left(\frac{11}{12}\right) e(712)e\left(\frac{7}{12}\right)
χ119(47,)\chi_{119}(47,\cdot) 1-1 11 e(16)e\left(\frac{1}{6}\right) e(112)e\left(\frac{1}{12}\right) e(13)e\left(\frac{1}{3}\right) e(512)e\left(\frac{5}{12}\right) ii 1-1 e(16)e\left(\frac{1}{6}\right) e(712)e\left(\frac{7}{12}\right) e(112)e\left(\frac{1}{12}\right) e(512)e\left(\frac{5}{12}\right)
χ119(89,)\chi_{119}(89,\cdot) 1-1 11 e(16)e\left(\frac{1}{6}\right) e(712)e\left(\frac{7}{12}\right) e(13)e\left(\frac{1}{3}\right) e(1112)e\left(\frac{11}{12}\right) i-i 1-1 e(16)e\left(\frac{1}{6}\right) e(112)e\left(\frac{1}{12}\right) e(712)e\left(\frac{7}{12}\right) e(1112)e\left(\frac{11}{12}\right)
χ119(115,)\chi_{119}(115,\cdot) 1-1 11 e(56)e\left(\frac{5}{6}\right) e(512)e\left(\frac{5}{12}\right) e(23)e\left(\frac{2}{3}\right) e(112)e\left(\frac{1}{12}\right) ii 1-1 e(56)e\left(\frac{5}{6}\right) e(1112)e\left(\frac{11}{12}\right) e(512)e\left(\frac{5}{12}\right) e(112)e\left(\frac{1}{12}\right)