Properties

Label 6047.18
Modulus $6047$
Conductor $6047$
Order $3023$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6047, base_ring=CyclotomicField(6046))
 
M = H._module
 
chi = DirichletCharacter(H, M([3452]))
 
pari: [g,chi] = znchar(Mod(18,6047))
 

Basic properties

Modulus: \(6047\)
Conductor: \(6047\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3023\)
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 6047.c

\(\chi_{6047}(2,\cdot)\) \(\chi_{6047}(3,\cdot)\) \(\chi_{6047}(4,\cdot)\) \(\chi_{6047}(6,\cdot)\) \(\chi_{6047}(7,\cdot)\) \(\chi_{6047}(8,\cdot)\) \(\chi_{6047}(9,\cdot)\) \(\chi_{6047}(11,\cdot)\) \(\chi_{6047}(12,\cdot)\) \(\chi_{6047}(14,\cdot)\) \(\chi_{6047}(16,\cdot)\) \(\chi_{6047}(18,\cdot)\) \(\chi_{6047}(21,\cdot)\) \(\chi_{6047}(22,\cdot)\) \(\chi_{6047}(23,\cdot)\) \(\chi_{6047}(24,\cdot)\) \(\chi_{6047}(25,\cdot)\) \(\chi_{6047}(27,\cdot)\) \(\chi_{6047}(28,\cdot)\) \(\chi_{6047}(32,\cdot)\) \(\chi_{6047}(33,\cdot)\) \(\chi_{6047}(36,\cdot)\) \(\chi_{6047}(37,\cdot)\) \(\chi_{6047}(41,\cdot)\) \(\chi_{6047}(42,\cdot)\) \(\chi_{6047}(43,\cdot)\) \(\chi_{6047}(44,\cdot)\) \(\chi_{6047}(46,\cdot)\) \(\chi_{6047}(47,\cdot)\) \(\chi_{6047}(48,\cdot)\) ...

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: $\Q(\zeta_{3023})$
Fixed field: Number field defined by a degree 3023 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{1726}{3023}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6047 }(18, a) \) \(1\)\(1\)\(e\left(\frac{2728}{3023}\right)\)\(e\left(\frac{57}{3023}\right)\)\(e\left(\frac{2433}{3023}\right)\)\(e\left(\frac{1726}{3023}\right)\)\(e\left(\frac{2785}{3023}\right)\)\(e\left(\frac{1304}{3023}\right)\)\(e\left(\frac{2138}{3023}\right)\)\(e\left(\frac{114}{3023}\right)\)\(e\left(\frac{1431}{3023}\right)\)\(e\left(\frac{1473}{3023}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6047 }(18,a) \;\) at \(\;a = \) e.g. 2