Properties

Label 703.w
Modulus $703$
Conductor $703$
Order $9$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(703, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,2]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(16,703))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(703\)
Conductor: \(703\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: 9.9.59654416235884558133761.2

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{703}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(1\)
\(\chi_{703}(44,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(1\)
\(\chi_{703}(157,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(1\)
\(\chi_{703}(256,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(1\)
\(\chi_{703}(403,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(1\)
\(\chi_{703}(530,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(1\)