Properties

Label 8470.bd
Modulus $8470$
Conductor $35$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,10,0]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(243,8470))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(8470\)
Conductor: \(35\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 35.k
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: \(\Q(\zeta_{12})\)
Fixed field: \(\Q(\zeta_{35})^+\)

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(9\) \(13\) \(17\) \(19\) \(23\) \(27\) \(29\) \(31\) \(37\)
\(\chi_{8470}(243,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{12}\right)\) \(i\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{8470}(1937,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{8470}(2663,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{12}\right)\) \(i\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{8470}(4357,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{12}\right)\) \(-i\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\)