Properties

Label 6003.605
Modulus $6003$
Conductor $6003$
Order $462$
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(462))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,399,264]))
 
pari: [g,chi] = znchar(Mod(605,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
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.dl

\(\chi_{6003}(20,\cdot)\) \(\chi_{6003}(65,\cdot)\) \(\chi_{6003}(74,\cdot)\) \(\chi_{6003}(83,\cdot)\) \(\chi_{6003}(194,\cdot)\) \(\chi_{6003}(227,\cdot)\) \(\chi_{6003}(281,\cdot)\) \(\chi_{6003}(401,\cdot)\) \(\chi_{6003}(488,\cdot)\) \(\chi_{6003}(500,\cdot)\) \(\chi_{6003}(596,\cdot)\) \(\chi_{6003}(605,\cdot)\) \(\chi_{6003}(632,\cdot)\) \(\chi_{6003}(770,\cdot)\) \(\chi_{6003}(779,\cdot)\) \(\chi_{6003}(803,\cdot)\) \(\chi_{6003}(848,\cdot)\) \(\chi_{6003}(866,\cdot)\) \(\chi_{6003}(893,\cdot)\) \(\chi_{6003}(977,\cdot)\) \(\chi_{6003}(1010,\cdot)\) \(\chi_{6003}(1022,\cdot)\) \(\chi_{6003}(1031,\cdot)\) \(\chi_{6003}(1040,\cdot)\) \(\chi_{6003}(1109,\cdot)\) \(\chi_{6003}(1118,\cdot)\) \(\chi_{6003}(1184,\cdot)\) \(\chi_{6003}(1238,\cdot)\) \(\chi_{6003}(1325,\cdot)\) \(\chi_{6003}(1328,\cdot)\) ...

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: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial (not computed)

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{19}{22}\right),e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(605, a) \) \(1\)\(1\)\(e\left(\frac{215}{462}\right)\)\(e\left(\frac{215}{231}\right)\)\(e\left(\frac{62}{231}\right)\)\(e\left(\frac{431}{462}\right)\)\(e\left(\frac{61}{154}\right)\)\(e\left(\frac{113}{154}\right)\)\(e\left(\frac{52}{231}\right)\)\(e\left(\frac{164}{231}\right)\)\(e\left(\frac{92}{231}\right)\)\(e\left(\frac{199}{231}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(605,a) \;\) at \(\;a = \) e.g. 2