Properties

Label 4729.4
Modulus $4729$
Conductor $4729$
Order $394$
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([53]))
 
pari: [g,chi] = znchar(Mod(4,4729))
 

Basic properties

Modulus: \(4729\)
Conductor: \(4729\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(394\)
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.j

\(\chi_{4729}(4,\cdot)\) \(\chi_{4729}(6,\cdot)\) \(\chi_{4729}(9,\cdot)\) \(\chi_{4729}(64,\cdot)\) \(\chi_{4729}(96,\cdot)\) \(\chi_{4729}(144,\cdot)\) \(\chi_{4729}(197,\cdot)\) \(\chi_{4729}(216,\cdot)\) \(\chi_{4729}(242,\cdot)\) \(\chi_{4729}(254,\cdot)\) \(\chi_{4729}(324,\cdot)\) \(\chi_{4729}(350,\cdot)\) \(\chi_{4729}(355,\cdot)\) \(\chi_{4729}(363,\cdot)\) \(\chi_{4729}(381,\cdot)\) \(\chi_{4729}(413,\cdot)\) \(\chi_{4729}(422,\cdot)\) \(\chi_{4729}(451,\cdot)\) \(\chi_{4729}(454,\cdot)\) \(\chi_{4729}(455,\cdot)\) \(\chi_{4729}(458,\cdot)\) \(\chi_{4729}(475,\cdot)\) \(\chi_{4729}(486,\cdot)\) \(\chi_{4729}(525,\cdot)\) \(\chi_{4729}(607,\cdot)\) \(\chi_{4729}(613,\cdot)\) \(\chi_{4729}(633,\cdot)\) \(\chi_{4729}(634,\cdot)\) \(\chi_{4729}(643,\cdot)\) \(\chi_{4729}(670,\cdot)\) ...

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 394 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{53}{394}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(4, a) \) \(1\)\(1\)\(e\left(\frac{153}{197}\right)\)\(e\left(\frac{171}{197}\right)\)\(e\left(\frac{109}{197}\right)\)\(e\left(\frac{87}{197}\right)\)\(e\left(\frac{127}{197}\right)\)\(e\left(\frac{13}{197}\right)\)\(e\left(\frac{65}{197}\right)\)\(e\left(\frac{145}{197}\right)\)\(e\left(\frac{43}{197}\right)\)\(e\left(\frac{143}{394}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4729 }(4,a) \;\) at \(\;a = \) e.g. 2