Properties

Label 6003.5396
Modulus $6003$
Conductor $6003$
Order $924$
Real no
Primitive yes
Minimal yes
Parity odd

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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([770,882,33]))
 
pari: [g,chi] = znchar(Mod(5396,6003))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.dr

\(\chi_{6003}(11,\cdot)\) \(\chi_{6003}(14,\cdot)\) \(\chi_{6003}(56,\cdot)\) \(\chi_{6003}(113,\cdot)\) \(\chi_{6003}(155,\cdot)\) \(\chi_{6003}(176,\cdot)\) \(\chi_{6003}(182,\cdot)\) \(\chi_{6003}(218,\cdot)\) \(\chi_{6003}(221,\cdot)\) \(\chi_{6003}(263,\cdot)\) \(\chi_{6003}(272,\cdot)\) \(\chi_{6003}(293,\cdot)\) \(\chi_{6003}(329,\cdot)\) \(\chi_{6003}(356,\cdot)\) \(\chi_{6003}(362,\cdot)\) \(\chi_{6003}(398,\cdot)\) \(\chi_{6003}(425,\cdot)\) \(\chi_{6003}(479,\cdot)\) \(\chi_{6003}(536,\cdot)\) \(\chi_{6003}(569,\cdot)\) \(\chi_{6003}(572,\cdot)\) \(\chi_{6003}(590,\cdot)\) \(\chi_{6003}(617,\cdot)\) \(\chi_{6003}(635,\cdot)\) \(\chi_{6003}(641,\cdot)\) \(\chi_{6003}(659,\cdot)\) \(\chi_{6003}(677,\cdot)\) \(\chi_{6003}(686,\cdot)\) \(\chi_{6003}(704,\cdot)\) \(\chi_{6003}(707,\cdot)\) ...

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

\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{21}{22}\right),e\left(\frac{1}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(5396, a) \) \(-1\)\(1\)\(e\left(\frac{719}{924}\right)\)\(e\left(\frac{257}{462}\right)\)\(e\left(\frac{419}{462}\right)\)\(e\left(\frac{415}{462}\right)\)\(e\left(\frac{103}{308}\right)\)\(e\left(\frac{211}{308}\right)\)\(e\left(\frac{293}{924}\right)\)\(e\left(\frac{311}{462}\right)\)\(e\left(\frac{625}{924}\right)\)\(e\left(\frac{26}{231}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(5396,a) \;\) at \(\;a = \) e.g. 2