Properties

Label 4889.26
Modulus $4889$
Conductor $4889$
Order $1222$
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(1222))
 
M = H._module
 
chi = DirichletCharacter(H, M([935]))
 
pari: [g,chi] = znchar(Mod(26,4889))
 

Basic properties

Modulus: \(4889\)
Conductor: \(4889\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1222\)
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.n

\(\chi_{4889}(4,\cdot)\) \(\chi_{4889}(5,\cdot)\) \(\chi_{4889}(18,\cdot)\) \(\chi_{4889}(19,\cdot)\) \(\chi_{4889}(26,\cdot)\) \(\chi_{4889}(41,\cdot)\) \(\chi_{4889}(42,\cdot)\) \(\chi_{4889}(47,\cdot)\) \(\chi_{4889}(59,\cdot)\) \(\chi_{4889}(64,\cdot)\) \(\chi_{4889}(80,\cdot)\) \(\chi_{4889}(81,\cdot)\) \(\chi_{4889}(88,\cdot)\) \(\chi_{4889}(98,\cdot)\) \(\chi_{4889}(100,\cdot)\) \(\chi_{4889}(106,\cdot)\) \(\chi_{4889}(115,\cdot)\) \(\chi_{4889}(117,\cdot)\) \(\chi_{4889}(121,\cdot)\) \(\chi_{4889}(125,\cdot)\) \(\chi_{4889}(129,\cdot)\) \(\chi_{4889}(134,\cdot)\) \(\chi_{4889}(169,\cdot)\) \(\chi_{4889}(174,\cdot)\) \(\chi_{4889}(189,\cdot)\) \(\chi_{4889}(218,\cdot)\) \(\chi_{4889}(269,\cdot)\) \(\chi_{4889}(273,\cdot)\) \(\chi_{4889}(288,\cdot)\) \(\chi_{4889}(292,\cdot)\) ...

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: $\Q(\zeta_{611})$
Fixed field: Number field defined by a degree 1222 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{935}{1222}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4889 }(26, a) \) \(1\)\(1\)\(e\left(\frac{111}{611}\right)\)\(e\left(\frac{935}{1222}\right)\)\(e\left(\frac{222}{611}\right)\)\(e\left(\frac{38}{611}\right)\)\(e\left(\frac{89}{94}\right)\)\(e\left(\frac{255}{1222}\right)\)\(e\left(\frac{333}{611}\right)\)\(e\left(\frac{324}{611}\right)\)\(e\left(\frac{149}{611}\right)\)\(e\left(\frac{553}{611}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4889 }(26,a) \;\) at \(\;a = \) e.g. 2