Properties

Label 1872.31
Modulus 18721872
Conductor 468468
Order 1212
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,0,4,9]))
 
pari: [g,chi] = znchar(Mod(31,1872))
 

Basic properties

Modulus: 18721872
Conductor: 468468
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 1212
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from χ468(31,)\chi_{468}(31,\cdot)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1872.fg

χ1872(31,)\chi_{1872}(31,\cdot) χ1872(463,)\chi_{1872}(463,\cdot) χ1872(655,)\chi_{1872}(655,\cdot) χ1872(1087,)\chi_{1872}(1087,\cdot)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(ζ12)\Q(\zeta_{12})
Fixed field: 12.12.1869778640298827501568.1

Values on generators

(703,469,209,145)(703,469,209,145)(1,1,e(13),i)(-1,1,e\left(\frac{1}{3}\right),-i)

First values

aa 1-111557711111717191923232525292931313535
χ1872(31,a) \chi_{ 1872 }(31, a) 1111e(512)e\left(\frac{5}{12}\right)e(112)e\left(\frac{1}{12}\right)e(112)e\left(\frac{1}{12}\right)1-1iie(23)e\left(\frac{2}{3}\right)e(56)e\left(\frac{5}{6}\right)e(13)e\left(\frac{1}{3}\right)e(1112)e\left(\frac{11}{12}\right)1-1
sage: chi.jacobi_sum(n)
 
χ1872(31,a)   \chi_{ 1872 }(31,a) \; at   a=\;a = e.g. 2