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Properties

Label 8.b
Modulus $8$
Conductor $8$
Order $2$
Real yes
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([0,1])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(5,8)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Kronecker symbol representation

Copy content sage:kronecker_character(8)
 
Copy content pari:znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{8}{\bullet}\right)\)

Basic properties

Modulus: \(8\)
Conductor: \(8\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(2\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: yes
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{2}) \)

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\)
\(\chi_{8}(5,\cdot)\) \(1\) \(1\) \(-1\) \(-1\)