Properties

Label 2176.w
Modulus $2176$
Conductor $544$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2176, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,7,1]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(111,2176))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2176\)
Conductor: \(544\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 544.w
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.0.881195590409519104.17

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(19\) \(21\) \(23\)
\(\chi_{2176}(111,\cdot)\) \(-1\) \(1\) \(i\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{2176}(943,\cdot)\) \(-1\) \(1\) \(i\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(-1\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\)
\(\chi_{2176}(1039,\cdot)\) \(-1\) \(1\) \(-i\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(i\) \(e\left(\frac{3}{8}\right)\) \(i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\)
\(\chi_{2176}(1103,\cdot)\) \(-1\) \(1\) \(-i\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(i\) \(e\left(\frac{7}{8}\right)\) \(i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\)