Properties

Label 4889.8
Modulus $4889$
Conductor $4889$
Order $2444$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4889, base_ring=CyclotomicField(2444))
 
M = H._module
 
chi = DirichletCharacter(H, M([1851]))
 
pari: [g,chi] = znchar(Mod(8,4889))
 

Basic properties

Modulus: \(4889\)
Conductor: \(4889\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2444\)
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 4889.o

\(\chi_{4889}(2,\cdot)\) \(\chi_{4889}(8,\cdot)\) \(\chi_{4889}(9,\cdot)\) \(\chi_{4889}(10,\cdot)\) \(\chi_{4889}(11,\cdot)\) \(\chi_{4889}(13,\cdot)\) \(\chi_{4889}(21,\cdot)\) \(\chi_{4889}(32,\cdot)\) \(\chi_{4889}(38,\cdot)\) \(\chi_{4889}(40,\cdot)\) \(\chi_{4889}(44,\cdot)\) \(\chi_{4889}(45,\cdot)\) \(\chi_{4889}(46,\cdot)\) \(\chi_{4889}(49,\cdot)\) \(\chi_{4889}(50,\cdot)\) \(\chi_{4889}(51,\cdot)\) \(\chi_{4889}(52,\cdot)\) \(\chi_{4889}(53,\cdot)\) \(\chi_{4889}(55,\cdot)\) \(\chi_{4889}(61,\cdot)\) \(\chi_{4889}(65,\cdot)\) \(\chi_{4889}(83,\cdot)\) \(\chi_{4889}(84,\cdot)\) \(\chi_{4889}(87,\cdot)\) \(\chi_{4889}(94,\cdot)\) \(\chi_{4889}(105,\cdot)\) \(\chi_{4889}(109,\cdot)\) \(\chi_{4889}(111,\cdot)\) \(\chi_{4889}(118,\cdot)\) \(\chi_{4889}(128,\cdot)\) ...

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: $\Q(\zeta_{2444})$
Fixed field: Number field defined by a degree 2444 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1851}{2444}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4889 }(8, a) \) \(1\)\(1\)\(e\left(\frac{719}{1222}\right)\)\(e\left(\frac{1851}{2444}\right)\)\(e\left(\frac{108}{611}\right)\)\(e\left(\frac{35}{611}\right)\)\(e\left(\frac{65}{188}\right)\)\(e\left(\frac{727}{2444}\right)\)\(e\left(\frac{935}{1222}\right)\)\(e\left(\frac{629}{1222}\right)\)\(e\left(\frac{789}{1222}\right)\)\(e\left(\frac{1083}{1222}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4889 }(8,a) \;\) at \(\;a = \) e.g. 2