Properties

Label 6041.5
Modulus $6041$
Conductor $6041$
Order $2586$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6041, base_ring=CyclotomicField(2586))
 
M = H._module
 
chi = DirichletCharacter(H, M([2155,3]))
 
pari: [g,chi] = znchar(Mod(5,6041))
 

Basic properties

Modulus: \(6041\)
Conductor: \(6041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2586\)
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 6041.o

\(\chi_{6041}(5,\cdot)\) \(\chi_{6041}(10,\cdot)\) \(\chi_{6041}(26,\cdot)\) \(\chi_{6041}(33,\cdot)\) \(\chi_{6041}(40,\cdot)\) \(\chi_{6041}(45,\cdot)\) \(\chi_{6041}(47,\cdot)\) \(\chi_{6041}(52,\cdot)\) \(\chi_{6041}(66,\cdot)\) \(\chi_{6041}(73,\cdot)\) \(\chi_{6041}(80,\cdot)\) \(\chi_{6041}(89,\cdot)\) \(\chi_{6041}(94,\cdot)\) \(\chi_{6041}(101,\cdot)\) \(\chi_{6041}(117,\cdot)\) \(\chi_{6041}(131,\cdot)\) \(\chi_{6041}(138,\cdot)\) \(\chi_{6041}(145,\cdot)\) \(\chi_{6041}(157,\cdot)\) \(\chi_{6041}(166,\cdot)\) \(\chi_{6041}(173,\cdot)\) \(\chi_{6041}(178,\cdot)\) \(\chi_{6041}(180,\cdot)\) \(\chi_{6041}(185,\cdot)\) \(\chi_{6041}(187,\cdot)\) \(\chi_{6041}(194,\cdot)\) \(\chi_{6041}(201,\cdot)\) \(\chi_{6041}(208,\cdot)\) \(\chi_{6041}(215,\cdot)\) \(\chi_{6041}(227,\cdot)\) ...

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: $\Q(\zeta_{1293})$
Fixed field: Number field defined by a degree 2586 polynomial (not computed)

Values on generators

\((864,4320)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{862}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 6041 }(5, a) \) \(1\)\(1\)\(e\left(\frac{457}{1293}\right)\)\(e\left(\frac{919}{2586}\right)\)\(e\left(\frac{914}{1293}\right)\)\(e\left(\frac{217}{1293}\right)\)\(e\left(\frac{611}{862}\right)\)\(e\left(\frac{26}{431}\right)\)\(e\left(\frac{919}{1293}\right)\)\(e\left(\frac{674}{1293}\right)\)\(e\left(\frac{1321}{2586}\right)\)\(e\left(\frac{161}{2586}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6041 }(5,a) \;\) at \(\;a = \) e.g. 2