Properties

Label 6003.619
Modulus $6003$
Conductor $6003$
Order $924$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(924))
 
M = H._module
 
chi = DirichletCharacter(H, M([616,546,759]))
 
pari: [g,chi] = znchar(Mod(619,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(924\)
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 6003.dp

\(\chi_{6003}(40,\cdot)\) \(\chi_{6003}(43,\cdot)\) \(\chi_{6003}(61,\cdot)\) \(\chi_{6003}(76,\cdot)\) \(\chi_{6003}(79,\cdot)\) \(\chi_{6003}(97,\cdot)\) \(\chi_{6003}(106,\cdot)\) \(\chi_{6003}(130,\cdot)\) \(\chi_{6003}(148,\cdot)\) \(\chi_{6003}(166,\cdot)\) \(\chi_{6003}(205,\cdot)\) \(\chi_{6003}(214,\cdot)\) \(\chi_{6003}(247,\cdot)\) \(\chi_{6003}(250,\cdot)\) \(\chi_{6003}(304,\cdot)\) \(\chi_{6003}(337,\cdot)\) \(\chi_{6003}(385,\cdot)\) \(\chi_{6003}(421,\cdot)\) \(\chi_{6003}(454,\cdot)\) \(\chi_{6003}(475,\cdot)\) \(\chi_{6003}(490,\cdot)\) \(\chi_{6003}(511,\cdot)\) \(\chi_{6003}(520,\cdot)\) \(\chi_{6003}(562,\cdot)\) \(\chi_{6003}(619,\cdot)\) \(\chi_{6003}(628,\cdot)\) \(\chi_{6003}(664,\cdot)\) \(\chi_{6003}(682,\cdot)\) \(\chi_{6003}(688,\cdot)\) \(\chi_{6003}(727,\cdot)\) ...

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: $\Q(\zeta_{924})$
Fixed field: Number field defined by a degree 924 polynomial (not computed)

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{13}{22}\right),e\left(\frac{23}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(619, a) \) \(1\)\(1\)\(e\left(\frac{619}{924}\right)\)\(e\left(\frac{157}{462}\right)\)\(e\left(\frac{230}{231}\right)\)\(e\left(\frac{347}{462}\right)\)\(e\left(\frac{3}{308}\right)\)\(e\left(\frac{205}{308}\right)\)\(e\left(\frac{481}{924}\right)\)\(e\left(\frac{181}{462}\right)\)\(e\left(\frac{389}{924}\right)\)\(e\left(\frac{157}{231}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(619,a) \;\) at \(\;a = \) e.g. 2