Properties

Label 6043.9
Modulus $6043$
Conductor $6043$
Order $1007$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6043, base_ring=CyclotomicField(2014))
 
M = H._module
 
chi = DirichletCharacter(H, M([1958]))
 
pari: [g,chi] = znchar(Mod(9,6043))
 

Basic properties

Modulus: \(6043\)
Conductor: \(6043\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1007\)
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 6043.m

\(\chi_{6043}(9,\cdot)\) \(\chi_{6043}(14,\cdot)\) \(\chi_{6043}(23,\cdot)\) \(\chi_{6043}(24,\cdot)\) \(\chi_{6043}(26,\cdot)\) \(\chi_{6043}(35,\cdot)\) \(\chi_{6043}(37,\cdot)\) \(\chi_{6043}(38,\cdot)\) \(\chi_{6043}(41,\cdot)\) \(\chi_{6043}(53,\cdot)\) \(\chi_{6043}(60,\cdot)\) \(\chi_{6043}(65,\cdot)\) \(\chi_{6043}(67,\cdot)\) \(\chi_{6043}(77,\cdot)\) \(\chi_{6043}(81,\cdot)\) \(\chi_{6043}(83,\cdot)\) \(\chi_{6043}(94,\cdot)\) \(\chi_{6043}(95,\cdot)\) \(\chi_{6043}(102,\cdot)\) \(\chi_{6043}(107,\cdot)\) \(\chi_{6043}(118,\cdot)\) \(\chi_{6043}(124,\cdot)\) \(\chi_{6043}(126,\cdot)\) \(\chi_{6043}(132,\cdot)\) \(\chi_{6043}(143,\cdot)\) \(\chi_{6043}(150,\cdot)\) \(\chi_{6043}(160,\cdot)\) \(\chi_{6043}(181,\cdot)\) \(\chi_{6043}(194,\cdot)\) \(\chi_{6043}(196,\cdot)\) ...

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: $\Q(\zeta_{1007})$
Fixed field: Number field defined by a degree 1007 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{979}{1007}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6043 }(9, a) \) \(1\)\(1\)\(e\left(\frac{49}{53}\right)\)\(e\left(\frac{338}{1007}\right)\)\(e\left(\frac{45}{53}\right)\)\(e\left(\frac{979}{1007}\right)\)\(e\left(\frac{262}{1007}\right)\)\(e\left(\frac{345}{1007}\right)\)\(e\left(\frac{41}{53}\right)\)\(e\left(\frac{676}{1007}\right)\)\(e\left(\frac{903}{1007}\right)\)\(e\left(\frac{14}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6043 }(9,a) \;\) at \(\;a = \) e.g. 2