Properties

Label 3024.569
Modulus $3024$
Conductor $1512$
Order $18$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,1,6]))
 
pari: [g,chi] = znchar(Mod(569,3024))
 

Basic properties

Modulus: \(3024\)
Conductor: \(1512\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1512}(1325,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3024.fx

\(\chi_{3024}(473,\cdot)\) \(\chi_{3024}(569,\cdot)\) \(\chi_{3024}(1481,\cdot)\) \(\chi_{3024}(1577,\cdot)\) \(\chi_{3024}(2489,\cdot)\) \(\chi_{3024}(2585,\cdot)\)

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: Number field defined by a degree 18 polynomial

Values on generators

\((1135,757,785,2593)\) → \((1,-1,e\left(\frac{1}{18}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3024 }(569, a) \) \(-1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3024 }(569,a) \;\) at \(\;a = \) e.g. 2