Properties

Label 6047.10
Modulus $6047$
Conductor $6047$
Order $6046$
Real no
Primitive yes
Minimal yes
Parity odd

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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([1719]))
 
pari: [g,chi] = znchar(Mod(10,6047))
 

Basic properties

Modulus: \(6047\)
Conductor: \(6047\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6046\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6047.d

\(\chi_{6047}(5,\cdot)\) \(\chi_{6047}(10,\cdot)\) \(\chi_{6047}(13,\cdot)\) \(\chi_{6047}(15,\cdot)\) \(\chi_{6047}(17,\cdot)\) \(\chi_{6047}(19,\cdot)\) \(\chi_{6047}(20,\cdot)\) \(\chi_{6047}(26,\cdot)\) \(\chi_{6047}(29,\cdot)\) \(\chi_{6047}(30,\cdot)\) \(\chi_{6047}(31,\cdot)\) \(\chi_{6047}(34,\cdot)\) \(\chi_{6047}(35,\cdot)\) \(\chi_{6047}(38,\cdot)\) \(\chi_{6047}(39,\cdot)\) \(\chi_{6047}(40,\cdot)\) \(\chi_{6047}(45,\cdot)\) \(\chi_{6047}(51,\cdot)\) \(\chi_{6047}(52,\cdot)\) \(\chi_{6047}(53,\cdot)\) \(\chi_{6047}(55,\cdot)\) \(\chi_{6047}(57,\cdot)\) \(\chi_{6047}(58,\cdot)\) \(\chi_{6047}(59,\cdot)\) \(\chi_{6047}(60,\cdot)\) \(\chi_{6047}(61,\cdot)\) \(\chi_{6047}(62,\cdot)\) \(\chi_{6047}(67,\cdot)\) \(\chi_{6047}(68,\cdot)\) \(\chi_{6047}(70,\cdot)\) ...

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 6046 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{1719}{6046}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 6047 }(10, a) \) \(-1\)\(1\)\(e\left(\frac{1397}{3023}\right)\)\(e\left(\frac{17}{3023}\right)\)\(e\left(\frac{2794}{3023}\right)\)\(e\left(\frac{1719}{6046}\right)\)\(e\left(\frac{1414}{3023}\right)\)\(e\left(\frac{2033}{3023}\right)\)\(e\left(\frac{1168}{3023}\right)\)\(e\left(\frac{34}{3023}\right)\)\(e\left(\frac{4513}{6046}\right)\)\(e\left(\frac{2985}{3023}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6047 }(10,a) \;\) at \(\;a = \) e.g. 2