Properties

Label 4729.111
Modulus $4729$
Conductor $4729$
Order $2364$
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(2364))
 
M = H._module
 
chi = DirichletCharacter(H, M([407]))
 
pari: [g,chi] = znchar(Mod(111,4729))
 

Basic properties

Modulus: \(4729\)
Conductor: \(4729\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2364\)
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.o

\(\chi_{4729}(5,\cdot)\) \(\chi_{4729}(7,\cdot)\) \(\chi_{4729}(20,\cdot)\) \(\chi_{4729}(26,\cdot)\) \(\chi_{4729}(28,\cdot)\) \(\chi_{4729}(30,\cdot)\) \(\chi_{4729}(38,\cdot)\) \(\chi_{4729}(39,\cdot)\) \(\chi_{4729}(42,\cdot)\) \(\chi_{4729}(45,\cdot)\) \(\chi_{4729}(50,\cdot)\) \(\chi_{4729}(57,\cdot)\) \(\chi_{4729}(59,\cdot)\) \(\chi_{4729}(63,\cdot)\) \(\chi_{4729}(65,\cdot)\) \(\chi_{4729}(70,\cdot)\) \(\chi_{4729}(71,\cdot)\) \(\chi_{4729}(73,\cdot)\) \(\chi_{4729}(74,\cdot)\) \(\chi_{4729}(75,\cdot)\) \(\chi_{4729}(80,\cdot)\) \(\chi_{4729}(91,\cdot)\) \(\chi_{4729}(95,\cdot)\) \(\chi_{4729}(98,\cdot)\) \(\chi_{4729}(104,\cdot)\) \(\chi_{4729}(105,\cdot)\) \(\chi_{4729}(111,\cdot)\) \(\chi_{4729}(112,\cdot)\) \(\chi_{4729}(120,\cdot)\) \(\chi_{4729}(133,\cdot)\) ...

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: $\Q(\zeta_{2364})$
Fixed field: Number field defined by a degree 2364 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{407}{2364}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4729 }(111, a) \) \(1\)\(1\)\(e\left(\frac{295}{394}\right)\)\(e\left(\frac{237}{394}\right)\)\(e\left(\frac{98}{197}\right)\)\(e\left(\frac{1129}{1182}\right)\)\(e\left(\frac{69}{197}\right)\)\(e\left(\frac{925}{1182}\right)\)\(e\left(\frac{97}{394}\right)\)\(e\left(\frac{40}{197}\right)\)\(e\left(\frac{416}{591}\right)\)\(e\left(\frac{371}{788}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4729 }(111,a) \;\) at \(\;a = \) e.g. 2