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Properties

Label 8112.ce
Modulus $8112$
Conductor $624$
Order $12$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([0,3,6,5])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(5765,8112)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(8112\)
Conductor: \(624\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 624.ce
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.11222620070536916307542016.1

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{8112}(5765,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{11}{12}\right)\) \(-i\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{8112}(6173,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{1}{12}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{8112}(7685,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{8112}(8093,\cdot)\) \(1\) \(1\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{5}{12}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\)