Properties

Label 6008.491
Modulus $6008$
Conductor $6008$
Order $250$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6008, base_ring=CyclotomicField(250))
 
M = H._module
 
chi = DirichletCharacter(H, M([125,125,67]))
 
pari: [g,chi] = znchar(Mod(491,6008))
 

Basic properties

Modulus: \(6008\)
Conductor: \(6008\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(250\)
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 6008.ca

\(\chi_{6008}(27,\cdot)\) \(\chi_{6008}(75,\cdot)\) \(\chi_{6008}(99,\cdot)\) \(\chi_{6008}(155,\cdot)\) \(\chi_{6008}(267,\cdot)\) \(\chi_{6008}(275,\cdot)\) \(\chi_{6008}(315,\cdot)\) \(\chi_{6008}(323,\cdot)\) \(\chi_{6008}(363,\cdot)\) \(\chi_{6008}(371,\cdot)\) \(\chi_{6008}(395,\cdot)\) \(\chi_{6008}(491,\cdot)\) \(\chi_{6008}(507,\cdot)\) \(\chi_{6008}(603,\cdot)\) \(\chi_{6008}(619,\cdot)\) \(\chi_{6008}(651,\cdot)\) \(\chi_{6008}(715,\cdot)\) \(\chi_{6008}(811,\cdot)\) \(\chi_{6008}(819,\cdot)\) \(\chi_{6008}(875,\cdot)\) \(\chi_{6008}(907,\cdot)\) \(\chi_{6008}(971,\cdot)\) \(\chi_{6008}(979,\cdot)\) \(\chi_{6008}(1067,\cdot)\) \(\chi_{6008}(1083,\cdot)\) \(\chi_{6008}(1155,\cdot)\) \(\chi_{6008}(1315,\cdot)\) \(\chi_{6008}(1459,\cdot)\) \(\chi_{6008}(1659,\cdot)\) \(\chi_{6008}(1675,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 250 polynomial (not computed)

Values on generators

\((1503,3005,1505)\) → \((-1,-1,e\left(\frac{67}{250}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6008 }(491, a) \) \(1\)\(1\)\(e\left(\frac{67}{250}\right)\)\(e\left(\frac{187}{250}\right)\)\(e\left(\frac{121}{125}\right)\)\(e\left(\frac{67}{125}\right)\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{161}{250}\right)\)\(e\left(\frac{2}{125}\right)\)\(e\left(\frac{43}{250}\right)\)\(e\left(\frac{103}{125}\right)\)\(e\left(\frac{59}{250}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6008 }(491,a) \;\) at \(\;a = \) e.g. 2