Properties

Label 6003.1397
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([77,147,363]))
 
pari: [g,chi] = znchar(Mod(1397,6003))
 

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.dh

\(\chi_{6003}(5,\cdot)\) \(\chi_{6003}(38,\cdot)\) \(\chi_{6003}(122,\cdot)\) \(\chi_{6003}(149,\cdot)\) \(\chi_{6003}(158,\cdot)\) \(\chi_{6003}(212,\cdot)\) \(\chi_{6003}(245,\cdot)\) \(\chi_{6003}(383,\cdot)\) \(\chi_{6003}(410,\cdot)\) \(\chi_{6003}(419,\cdot)\) \(\chi_{6003}(428,\cdot)\) \(\chi_{6003}(470,\cdot)\) \(\chi_{6003}(497,\cdot)\) \(\chi_{6003}(527,\cdot)\) \(\chi_{6003}(734,\cdot)\) \(\chi_{6003}(776,\cdot)\) \(\chi_{6003}(941,\cdot)\) \(\chi_{6003}(950,\cdot)\) \(\chi_{6003}(1019,\cdot)\) \(\chi_{6003}(1049,\cdot)\) \(\chi_{6003}(1193,\cdot)\) \(\chi_{6003}(1211,\cdot)\) \(\chi_{6003}(1253,\cdot)\) \(\chi_{6003}(1256,\cdot)\) \(\chi_{6003}(1280,\cdot)\) \(\chi_{6003}(1298,\cdot)\) \(\chi_{6003}(1397,\cdot)\) \(\chi_{6003}(1454,\cdot)\) \(\chi_{6003}(1463,\cdot)\) \(\chi_{6003}(1514,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{22}\right),e\left(\frac{11}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(1397, a) \) \(1\)\(1\)\(e\left(\frac{136}{231}\right)\)\(e\left(\frac{41}{231}\right)\)\(e\left(\frac{101}{231}\right)\)\(e\left(\frac{65}{462}\right)\)\(e\left(\frac{59}{77}\right)\)\(e\left(\frac{2}{77}\right)\)\(e\left(\frac{311}{462}\right)\)\(e\left(\frac{215}{231}\right)\)\(e\left(\frac{337}{462}\right)\)\(e\left(\frac{82}{231}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(1397,a) \;\) at \(\;a = \) e.g. 2