Properties

Label 4729.150
Modulus $4729$
Conductor $4729$
Order $591$
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([224]))
 
pari: [g,chi] = znchar(Mod(150,4729))
 

Basic properties

Modulus: \(4729\)
Conductor: \(4729\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(591\)
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.k

\(\chi_{4729}(14,\cdot)\) \(\chi_{4729}(19,\cdot)\) \(\chi_{4729}(21,\cdot)\) \(\chi_{4729}(35,\cdot)\) \(\chi_{4729}(37,\cdot)\) \(\chi_{4729}(40,\cdot)\) \(\chi_{4729}(52,\cdot)\) \(\chi_{4729}(60,\cdot)\) \(\chi_{4729}(67,\cdot)\) \(\chi_{4729}(78,\cdot)\) \(\chi_{4729}(83,\cdot)\) \(\chi_{4729}(90,\cdot)\) \(\chi_{4729}(100,\cdot)\) \(\chi_{4729}(117,\cdot)\) \(\chi_{4729}(118,\cdot)\) \(\chi_{4729}(130,\cdot)\) \(\chi_{4729}(131,\cdot)\) \(\chi_{4729}(135,\cdot)\) \(\chi_{4729}(150,\cdot)\) \(\chi_{4729}(169,\cdot)\) \(\chi_{4729}(177,\cdot)\) \(\chi_{4729}(187,\cdot)\) \(\chi_{4729}(193,\cdot)\) \(\chi_{4729}(195,\cdot)\) \(\chi_{4729}(196,\cdot)\) \(\chi_{4729}(224,\cdot)\) \(\chi_{4729}(225,\cdot)\) \(\chi_{4729}(266,\cdot)\) \(\chi_{4729}(294,\cdot)\) \(\chi_{4729}(304,\cdot)\) ...

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 591 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{112}{591}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(150, a) \) \(1\)\(1\)\(e\left(\frac{52}{197}\right)\)\(e\left(\frac{174}{197}\right)\)\(e\left(\frac{104}{197}\right)\)\(e\left(\frac{193}{591}\right)\)\(e\left(\frac{29}{197}\right)\)\(e\left(\frac{133}{591}\right)\)\(e\left(\frac{156}{197}\right)\)\(e\left(\frac{151}{197}\right)\)\(e\left(\frac{349}{591}\right)\)\(e\left(\frac{14}{197}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4729 }(150,a) \;\) at \(\;a = \) e.g. 2