Properties

Label 4003.993
Modulus $4003$
Conductor $4003$
Order $69$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4003, base_ring=CyclotomicField(138))
 
M = H._module
 
chi = DirichletCharacter(H, M([124]))
 
pari: [g,chi] = znchar(Mod(993,4003))
 

Basic properties

Modulus: \(4003\)
Conductor: \(4003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(69\)
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.i

\(\chi_{4003}(129,\cdot)\) \(\chi_{4003}(493,\cdot)\) \(\chi_{4003}(534,\cdot)\) \(\chi_{4003}(583,\cdot)\) \(\chi_{4003}(629,\cdot)\) \(\chi_{4003}(651,\cdot)\) \(\chi_{4003}(943,\cdot)\) \(\chi_{4003}(993,\cdot)\) \(\chi_{4003}(1183,\cdot)\) \(\chi_{4003}(1265,\cdot)\) \(\chi_{4003}(1311,\cdot)\) \(\chi_{4003}(1434,\cdot)\) \(\chi_{4003}(1543,\cdot)\) \(\chi_{4003}(1557,\cdot)\) \(\chi_{4003}(1752,\cdot)\) \(\chi_{4003}(1821,\cdot)\) \(\chi_{4003}(1825,\cdot)\) \(\chi_{4003}(1857,\cdot)\) \(\chi_{4003}(1866,\cdot)\) \(\chi_{4003}(1914,\cdot)\) \(\chi_{4003}(2015,\cdot)\) \(\chi_{4003}(2321,\cdot)\) \(\chi_{4003}(2434,\cdot)\) \(\chi_{4003}(2442,\cdot)\) \(\chi_{4003}(2487,\cdot)\) \(\chi_{4003}(2621,\cdot)\) \(\chi_{4003}(2735,\cdot)\) \(\chi_{4003}(2817,\cdot)\) \(\chi_{4003}(2869,\cdot)\) \(\chi_{4003}(2897,\cdot)\) ...

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: $\Q(\zeta_{69})$
Fixed field: Number field defined by a degree 69 polynomial

Values on generators

\(2\) → \(e\left(\frac{62}{69}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4003 }(993, a) \) \(1\)\(1\)\(e\left(\frac{62}{69}\right)\)\(e\left(\frac{55}{69}\right)\)\(e\left(\frac{55}{69}\right)\)\(e\left(\frac{53}{69}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{35}{69}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{41}{69}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{22}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4003 }(993,a) \;\) at \(\;a = \) e.g. 2