Properties

Label 273.bs
Modulus $273$
Conductor $273$
Order $12$
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([6,2,11]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(59,273))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(273\)
Conductor: \(273\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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: \(\Q(\zeta_{12})\)
Fixed field: 12.0.369049655140530434165277.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(11\) \(16\) \(17\) \(19\) \(20\)
\(\chi_{273}(59,\cdot)\) \(-1\) \(1\) \(-i\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(1\) \(-1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{273}(89,\cdot)\) \(-1\) \(1\) \(-i\) \(-1\) \(e\left(\frac{11}{12}\right)\) \(i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{273}(227,\cdot)\) \(-1\) \(1\) \(i\) \(-1\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(1\) \(-1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{273}(236,\cdot)\) \(-1\) \(1\) \(i\) \(-1\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(1\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\)