Properties

Label 273.20
Modulus 273273
Conductor 273273
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(273, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,6,11]))
 
pari: [g,chi] = znchar(Mod(20,273))
 

Basic properties

Modulus: 273273
Conductor: 273273
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)
 

Galois orbit 273.ca

χ273(20,)\chi_{273}(20,\cdot) χ273(41,)\chi_{273}(41,\cdot) χ273(167,)\chi_{273}(167,\cdot) χ273(188,)\chi_{273}(188,\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: 12.0.153706645206385020477.1

Values on generators

(92,157,106)(92,157,106)(1,1,e(1112))(-1,-1,e\left(\frac{11}{12}\right))

First values

aa 1-11122445588101011111616171719192020
χ273(20,a) \chi_{ 273 }(20, a) 1-111e(512)e\left(\frac{5}{12}\right)e(56)e\left(\frac{5}{6}\right)iiiie(23)e\left(\frac{2}{3}\right)e(1112)e\left(\frac{11}{12}\right)e(23)e\left(\frac{2}{3}\right)e(56)e\left(\frac{5}{6}\right)e(112)e\left(\frac{1}{12}\right)e(112)e\left(\frac{1}{12}\right)
sage: chi.jacobi_sum(n)
 
χ273(20,a)   \chi_{ 273 }(20,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ273(20,))   \tau_{ a }( \chi_{ 273 }(20,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ273(20,),χ273(n,))   J(\chi_{ 273 }(20,·),\chi_{ 273 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ273(20,))  K(a,b,\chi_{ 273 }(20,·)) \; at   a,b=\; a,b = e.g. 1,2