Properties

Label 576.x
Modulus $576$
Conductor $96$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(576, base_ring=CyclotomicField(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1,4]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(89,576))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(576\)
Conductor: \(96\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 96.p
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: 8.0.173946175488.1

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{576}(89,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(e\left(\frac{7}{8}\right)\) \(i\) \(i\) \(e\left(\frac{7}{8}\right)\) \(1\)
\(\chi_{576}(233,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(e\left(\frac{1}{8}\right)\) \(-i\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(1\)
\(\chi_{576}(377,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(1\) \(e\left(\frac{3}{8}\right)\) \(i\) \(i\) \(e\left(\frac{3}{8}\right)\) \(1\)
\(\chi_{576}(521,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(-i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(1\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(1\)