Properties

Label 6008.1621
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([0,75,46]))
 
pari: [g,chi] = znchar(Mod(1621,6008))
 

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.br

\(\chi_{6008}(61,\cdot)\) \(\chi_{6008}(197,\cdot)\) \(\chi_{6008}(229,\cdot)\) \(\chi_{6008}(405,\cdot)\) \(\chi_{6008}(637,\cdot)\) \(\chi_{6008}(717,\cdot)\) \(\chi_{6008}(837,\cdot)\) \(\chi_{6008}(941,\cdot)\) \(\chi_{6008}(1125,\cdot)\) \(\chi_{6008}(1373,\cdot)\) \(\chi_{6008}(1621,\cdot)\) \(\chi_{6008}(1901,\cdot)\) \(\chi_{6008}(1973,\cdot)\) \(\chi_{6008}(2133,\cdot)\) \(\chi_{6008}(2141,\cdot)\) \(\chi_{6008}(2285,\cdot)\) \(\chi_{6008}(2437,\cdot)\) \(\chi_{6008}(2453,\cdot)\) \(\chi_{6008}(2549,\cdot)\) \(\chi_{6008}(2781,\cdot)\) \(\chi_{6008}(3125,\cdot)\) \(\chi_{6008}(3277,\cdot)\) \(\chi_{6008}(3885,\cdot)\) \(\chi_{6008}(3949,\cdot)\) \(\chi_{6008}(4133,\cdot)\) \(\chi_{6008}(4613,\cdot)\) \(\chi_{6008}(4637,\cdot)\) \(\chi_{6008}(4669,\cdot)\) \(\chi_{6008}(4805,\cdot)\) \(\chi_{6008}(4813,\cdot)\) ...

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

\((1503,3005,1505)\) → \((1,-1,e\left(\frac{23}{75}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6008 }(1621, a) \) \(1\)\(1\)\(e\left(\frac{121}{150}\right)\)\(e\left(\frac{31}{150}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{46}{75}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{43}{150}\right)\)\(e\left(\frac{1}{75}\right)\)\(e\left(\frac{67}{75}\right)\)\(e\left(\frac{103}{150}\right)\)\(e\left(\frac{67}{150}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6008 }(1621,a) \;\) at \(\;a = \) e.g. 2