Properties

Label 4729.319
Modulus $4729$
Conductor $4729$
Order $197$
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(394))
 
M = H._module
 
chi = DirichletCharacter(H, M([386]))
 
pari: [g,chi] = znchar(Mod(319,4729))
 

Basic properties

Modulus: \(4729\)
Conductor: \(4729\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(197\)
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.i

\(\chi_{4729}(16,\cdot)\) \(\chi_{4729}(24,\cdot)\) \(\chi_{4729}(36,\cdot)\) \(\chi_{4729}(54,\cdot)\) \(\chi_{4729}(81,\cdot)\) \(\chi_{4729}(250,\cdot)\) \(\chi_{4729}(251,\cdot)\) \(\chi_{4729}(253,\cdot)\) \(\chi_{4729}(256,\cdot)\) \(\chi_{4729}(295,\cdot)\) \(\chi_{4729}(314,\cdot)\) \(\chi_{4729}(319,\cdot)\) \(\chi_{4729}(325,\cdot)\) \(\chi_{4729}(375,\cdot)\) \(\chi_{4729}(384,\cdot)\) \(\chi_{4729}(471,\cdot)\) \(\chi_{4729}(490,\cdot)\) \(\chi_{4729}(497,\cdot)\) \(\chi_{4729}(576,\cdot)\) \(\chi_{4729}(637,\cdot)\) \(\chi_{4729}(665,\cdot)\) \(\chi_{4729}(682,\cdot)\) \(\chi_{4729}(734,\cdot)\) \(\chi_{4729}(735,\cdot)\) \(\chi_{4729}(739,\cdot)\) \(\chi_{4729}(746,\cdot)\) \(\chi_{4729}(788,\cdot)\) \(\chi_{4729}(821,\cdot)\) \(\chi_{4729}(857,\cdot)\) \(\chi_{4729}(864,\cdot)\) ...

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: $\Q(\zeta_{197})$
Fixed field: Number field defined by a degree 197 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{193}{197}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(319, a) \) \(1\)\(1\)\(e\left(\frac{107}{197}\right)\)\(e\left(\frac{108}{197}\right)\)\(e\left(\frac{17}{197}\right)\)\(e\left(\frac{169}{197}\right)\)\(e\left(\frac{18}{197}\right)\)\(e\left(\frac{143}{197}\right)\)\(e\left(\frac{124}{197}\right)\)\(e\left(\frac{19}{197}\right)\)\(e\left(\frac{79}{197}\right)\)\(e\left(\frac{97}{197}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4729 }(319,a) \;\) at \(\;a = \) e.g. 2