Properties

Label 8001.22
Modulus $8001$
Conductor $1143$
Order $9$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,0,2]))
 
pari: [g,chi] = znchar(Mod(22,8001))
 

Basic properties

Modulus: \(8001\)
Conductor: \(1143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1143}(22,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8001.ek

\(\chi_{8001}(22,\cdot)\) \(\chi_{8001}(484,\cdot)\) \(\chi_{8001}(988,\cdot)\) \(\chi_{8001}(2227,\cdot)\) \(\chi_{8001}(6133,\cdot)\) \(\chi_{8001}(6910,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: 9.9.35965394160281315137521.4

Values on generators

\((3557,1144,7750)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{1}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 8001 }(22, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8001 }(22,a) \;\) at \(\;a = \) e.g. 2