Properties

Label 6008.3227
Modulus $6008$
Conductor $6008$
Order $150$
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(150))
 
M = H._module
 
chi = DirichletCharacter(H, M([75,75,101]))
 
pari: [g,chi] = znchar(Mod(3227,6008))
 

Basic properties

Modulus: \(6008\)
Conductor: \(6008\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(150\)
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.bt

\(\chi_{6008}(11,\cdot)\) \(\chi_{6008}(83,\cdot)\) \(\chi_{6008}(243,\cdot)\) \(\chi_{6008}(251,\cdot)\) \(\chi_{6008}(379,\cdot)\) \(\chi_{6008}(467,\cdot)\) \(\chi_{6008}(571,\cdot)\) \(\chi_{6008}(699,\cdot)\) \(\chi_{6008}(859,\cdot)\) \(\chi_{6008}(1051,\cdot)\) \(\chi_{6008}(1195,\cdot)\) \(\chi_{6008}(1203,\cdot)\) \(\chi_{6008}(1339,\cdot)\) \(\chi_{6008}(1371,\cdot)\) \(\chi_{6008}(1395,\cdot)\) \(\chi_{6008}(1875,\cdot)\) \(\chi_{6008}(2059,\cdot)\) \(\chi_{6008}(2123,\cdot)\) \(\chi_{6008}(2731,\cdot)\) \(\chi_{6008}(2883,\cdot)\) \(\chi_{6008}(3227,\cdot)\) \(\chi_{6008}(3459,\cdot)\) \(\chi_{6008}(3555,\cdot)\) \(\chi_{6008}(3571,\cdot)\) \(\chi_{6008}(3723,\cdot)\) \(\chi_{6008}(3867,\cdot)\) \(\chi_{6008}(3875,\cdot)\) \(\chi_{6008}(4035,\cdot)\) \(\chi_{6008}(4107,\cdot)\) \(\chi_{6008}(4387,\cdot)\) ...

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: $\Q(\zeta_{75})$
Fixed field: Number field defined by a degree 150 polynomial (not computed)

Values on generators

\((1503,3005,1505)\) → \((-1,-1,e\left(\frac{101}{150}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6008 }(3227, a) \) \(1\)\(1\)\(e\left(\frac{101}{150}\right)\)\(e\left(\frac{11}{150}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{26}{75}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{83}{150}\right)\)\(e\left(\frac{56}{75}\right)\)\(e\left(\frac{79}{150}\right)\)\(e\left(\frac{34}{75}\right)\)\(e\left(\frac{77}{150}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6008 }(3227,a) \;\) at \(\;a = \) e.g. 2