Properties

Label 4729.23
Modulus $4729$
Conductor $4729$
Order $1576$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4729, base_ring=CyclotomicField(1576))
 
M = H._module
 
chi = DirichletCharacter(H, M([1259]))
 
pari: [g,chi] = znchar(Mod(23,4729))
 

Basic properties

Modulus: \(4729\)
Conductor: \(4729\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1576\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4729.n

\(\chi_{4729}(11,\cdot)\) \(\chi_{4729}(22,\cdot)\) \(\chi_{4729}(23,\cdot)\) \(\chi_{4729}(29,\cdot)\) \(\chi_{4729}(31,\cdot)\) \(\chi_{4729}(33,\cdot)\) \(\chi_{4729}(41,\cdot)\) \(\chi_{4729}(44,\cdot)\) \(\chi_{4729}(46,\cdot)\) \(\chi_{4729}(62,\cdot)\) \(\chi_{4729}(66,\cdot)\) \(\chi_{4729}(69,\cdot)\) \(\chi_{4729}(82,\cdot)\) \(\chi_{4729}(87,\cdot)\) \(\chi_{4729}(88,\cdot)\) \(\chi_{4729}(92,\cdot)\) \(\chi_{4729}(93,\cdot)\) \(\chi_{4729}(99,\cdot)\) \(\chi_{4729}(103,\cdot)\) \(\chi_{4729}(113,\cdot)\) \(\chi_{4729}(116,\cdot)\) \(\chi_{4729}(123,\cdot)\) \(\chi_{4729}(124,\cdot)\) \(\chi_{4729}(132,\cdot)\) \(\chi_{4729}(138,\cdot)\) \(\chi_{4729}(151,\cdot)\) \(\chi_{4729}(163,\cdot)\) \(\chi_{4729}(164,\cdot)\) \(\chi_{4729}(173,\cdot)\) \(\chi_{4729}(174,\cdot)\) ...

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: $\Q(\zeta_{1576})$
Fixed field: Number field defined by a degree 1576 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{1259}{1576}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(23, a) \) \(-1\)\(1\)\(e\left(\frac{29}{788}\right)\)\(e\left(\frac{241}{788}\right)\)\(e\left(\frac{29}{394}\right)\)\(e\left(\frac{565}{788}\right)\)\(e\left(\frac{135}{394}\right)\)\(e\left(\frac{175}{788}\right)\)\(e\left(\frac{87}{788}\right)\)\(e\left(\frac{241}{394}\right)\)\(e\left(\frac{297}{394}\right)\)\(e\left(\frac{349}{1576}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4729 }(23,a) \;\) at \(\;a = \) e.g. 2