Properties

Label 4003.103
Modulus $4003$
Conductor $4003$
Order $667$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4003, base_ring=CyclotomicField(1334))
 
M = H._module
 
chi = DirichletCharacter(H, M([1180]))
 
pari: [g,chi] = znchar(Mod(103,4003))
 

Basic properties

Modulus: \(4003\)
Conductor: \(4003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(667\)
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 4003.m

\(\chi_{4003}(6,\cdot)\) \(\chi_{4003}(11,\cdot)\) \(\chi_{4003}(13,\cdot)\) \(\chi_{4003}(15,\cdot)\) \(\chi_{4003}(17,\cdot)\) \(\chi_{4003}(28,\cdot)\) \(\chi_{4003}(36,\cdot)\) \(\chi_{4003}(64,\cdot)\) \(\chi_{4003}(66,\cdot)\) \(\chi_{4003}(69,\cdot)\) \(\chi_{4003}(70,\cdot)\) \(\chi_{4003}(73,\cdot)\) \(\chi_{4003}(76,\cdot)\) \(\chi_{4003}(78,\cdot)\) \(\chi_{4003}(86,\cdot)\) \(\chi_{4003}(90,\cdot)\) \(\chi_{4003}(93,\cdot)\) \(\chi_{4003}(94,\cdot)\) \(\chi_{4003}(103,\cdot)\) \(\chi_{4003}(121,\cdot)\) \(\chi_{4003}(142,\cdot)\) \(\chi_{4003}(143,\cdot)\) \(\chi_{4003}(149,\cdot)\) \(\chi_{4003}(151,\cdot)\) \(\chi_{4003}(160,\cdot)\) \(\chi_{4003}(165,\cdot)\) \(\chi_{4003}(169,\cdot)\) \(\chi_{4003}(175,\cdot)\) \(\chi_{4003}(183,\cdot)\) \(\chi_{4003}(187,\cdot)\) ...

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: $\Q(\zeta_{667})$
Fixed field: Number field defined by a degree 667 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{590}{667}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4003 }(103, a) \) \(1\)\(1\)\(e\left(\frac{590}{667}\right)\)\(e\left(\frac{651}{667}\right)\)\(e\left(\frac{513}{667}\right)\)\(e\left(\frac{123}{667}\right)\)\(e\left(\frac{574}{667}\right)\)\(e\left(\frac{431}{667}\right)\)\(e\left(\frac{436}{667}\right)\)\(e\left(\frac{635}{667}\right)\)\(e\left(\frac{2}{29}\right)\)\(e\left(\frac{243}{667}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4003 }(103,a) \;\) at \(\;a = \) e.g. 2