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Properties

Label 1309.x
Modulus $1309$
Conductor $187$
Order $8$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1309, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,4,1])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(43,1309)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1309\)
Conductor: \(187\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 187.i
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.0.6007768511393.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(8\) \(9\) \(10\) \(12\) \(13\)
\(\chi_{1309}(43,\cdot)\) \(-1\) \(1\) \(i\) \(e\left(\frac{1}{8}\right)\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(1\)
\(\chi_{1309}(274,\cdot)\) \(-1\) \(1\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(i\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(1\)
\(\chi_{1309}(967,\cdot)\) \(-1\) \(1\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(i\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(1\)
\(\chi_{1309}(1198,\cdot)\) \(-1\) \(1\) \(i\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(1\)