Properties

Label 4729.189
Modulus $4729$
Conductor $4729$
Order $1182$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4729, base_ring=CyclotomicField(1182))
 
M = H._module
 
chi = DirichletCharacter(H, M([451]))
 
pari: [g,chi] = znchar(Mod(189,4729))
 

Basic properties

Modulus: \(4729\)
Conductor: \(4729\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1182\)
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 4729.m

\(\chi_{4729}(10,\cdot)\) \(\chi_{4729}(13,\cdot)\) \(\chi_{4729}(15,\cdot)\) \(\chi_{4729}(25,\cdot)\) \(\chi_{4729}(49,\cdot)\) \(\chi_{4729}(56,\cdot)\) \(\chi_{4729}(76,\cdot)\) \(\chi_{4729}(84,\cdot)\) \(\chi_{4729}(97,\cdot)\) \(\chi_{4729}(114,\cdot)\) \(\chi_{4729}(126,\cdot)\) \(\chi_{4729}(140,\cdot)\) \(\chi_{4729}(142,\cdot)\) \(\chi_{4729}(146,\cdot)\) \(\chi_{4729}(148,\cdot)\) \(\chi_{4729}(149,\cdot)\) \(\chi_{4729}(160,\cdot)\) \(\chi_{4729}(171,\cdot)\) \(\chi_{4729}(179,\cdot)\) \(\chi_{4729}(182,\cdot)\) \(\chi_{4729}(189,\cdot)\) \(\chi_{4729}(190,\cdot)\) \(\chi_{4729}(208,\cdot)\) \(\chi_{4729}(210,\cdot)\) \(\chi_{4729}(213,\cdot)\) \(\chi_{4729}(219,\cdot)\) \(\chi_{4729}(222,\cdot)\) \(\chi_{4729}(240,\cdot)\) \(\chi_{4729}(241,\cdot)\) \(\chi_{4729}(247,\cdot)\) ...

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: $\Q(\zeta_{591})$
Fixed field: Number field defined by a degree 1182 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{451}{1182}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(189, a) \) \(1\)\(1\)\(e\left(\frac{66}{197}\right)\)\(e\left(\frac{39}{197}\right)\)\(e\left(\frac{132}{197}\right)\)\(e\left(\frac{101}{591}\right)\)\(e\left(\frac{105}{197}\right)\)\(e\left(\frac{434}{591}\right)\)\(e\left(\frac{1}{197}\right)\)\(e\left(\frac{78}{197}\right)\)\(e\left(\frac{299}{591}\right)\)\(e\left(\frac{81}{394}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4729 }(189,a) \;\) at \(\;a = \) e.g. 2