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Properties

Label 448.w
Modulus $448$
Conductor $32$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,5,0])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(71,448)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(448\)
Conductor: \(32\)
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 32.h
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
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: 8.0.2147483648.1

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(23\) \(25\)
\(\chi_{448}(71,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(i\) \(i\)
\(\chi_{448}(183,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(-i\)
\(\chi_{448}(295,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(i\) \(i\)
\(\chi_{448}(407,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(-i\)