Properties

Label 6004.2315
Modulus $6004$
Conductor $6004$
Order $18$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6004, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,4,3]))
 
pari: [g,chi] = znchar(Mod(2315,6004))
 

Basic properties

Modulus: \(6004\)
Conductor: \(6004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
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)
 

Galois orbit 6004.bw

\(\chi_{6004}(2315,\cdot)\) \(\chi_{6004}(4243,\cdot)\) \(\chi_{6004}(4527,\cdot)\) \(\chi_{6004}(4843,\cdot)\) \(\chi_{6004}(5191,\cdot)\) \(\chi_{6004}(5823,\cdot)\)

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

\((3003,2529,3953)\) → \((-1,e\left(\frac{2}{9}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 6004 }(2315, a) \) \(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{9}\right)\)\(-1\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6004 }(2315,a) \;\) at \(\;a = \) e.g. 2