Properties

Label 6043.6
Modulus $6043$
Conductor $6043$
Order $3021$
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(6042))
 
M = H._module
 
chi = DirichletCharacter(H, M([4666]))
 
pari: [g,chi] = znchar(Mod(6,6043))
 

Basic properties

Modulus: \(6043\)
Conductor: \(6043\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3021\)
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.o

\(\chi_{6043}(6,\cdot)\) \(\chi_{6043}(10,\cdot)\) \(\chi_{6043}(15,\cdot)\) \(\chi_{6043}(17,\cdot)\) \(\chi_{6043}(21,\cdot)\) \(\chi_{6043}(22,\cdot)\) \(\chi_{6043}(25,\cdot)\) \(\chi_{6043}(31,\cdot)\) \(\chi_{6043}(33,\cdot)\) \(\chi_{6043}(36,\cdot)\) \(\chi_{6043}(39,\cdot)\) \(\chi_{6043}(40,\cdot)\) \(\chi_{6043}(49,\cdot)\) \(\chi_{6043}(54,\cdot)\) \(\chi_{6043}(55,\cdot)\) \(\chi_{6043}(56,\cdot)\) \(\chi_{6043}(57,\cdot)\) \(\chi_{6043}(58,\cdot)\) \(\chi_{6043}(61,\cdot)\) \(\chi_{6043}(68,\cdot)\) \(\chi_{6043}(73,\cdot)\) \(\chi_{6043}(79,\cdot)\) \(\chi_{6043}(84,\cdot)\) \(\chi_{6043}(86,\cdot)\) \(\chi_{6043}(87,\cdot)\) \(\chi_{6043}(88,\cdot)\) \(\chi_{6043}(89,\cdot)\) \(\chi_{6043}(90,\cdot)\) \(\chi_{6043}(92,\cdot)\) \(\chi_{6043}(96,\cdot)\) ...

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: $\Q(\zeta_{3021})$
Fixed field: Number field defined by a degree 3021 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{2333}{3021}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6043 }(6, a) \) \(1\)\(1\)\(e\left(\frac{38}{159}\right)\)\(e\left(\frac{131}{1007}\right)\)\(e\left(\frac{76}{159}\right)\)\(e\left(\frac{2333}{3021}\right)\)\(e\left(\frac{1115}{3021}\right)\)\(e\left(\frac{565}{3021}\right)\)\(e\left(\frac{38}{53}\right)\)\(e\left(\frac{262}{1007}\right)\)\(e\left(\frac{34}{3021}\right)\)\(e\left(\frac{40}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6043 }(6,a) \;\) at \(\;a = \) e.g. 2