Properties

Label 2003.184
Modulus $2003$
Conductor $2003$
Order $143$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2003, base_ring=CyclotomicField(286))
 
M = H._module
 
chi = DirichletCharacter(H, M([60]))
 
pari: [g,chi] = znchar(Mod(184,2003))
 

Basic properties

Modulus: \(2003\)
Conductor: \(2003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(143\)
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 2003.k

\(\chi_{2003}(4,\cdot)\) \(\chi_{2003}(16,\cdot)\) \(\chi_{2003}(42,\cdot)\) \(\chi_{2003}(46,\cdot)\) \(\chi_{2003}(64,\cdot)\) \(\chi_{2003}(88,\cdot)\) \(\chi_{2003}(90,\cdot)\) \(\chi_{2003}(113,\cdot)\) \(\chi_{2003}(121,\cdot)\) \(\chi_{2003}(134,\cdot)\) \(\chi_{2003}(141,\cdot)\) \(\chi_{2003}(142,\cdot)\) \(\chi_{2003}(155,\cdot)\) \(\chi_{2003}(168,\cdot)\) \(\chi_{2003}(174,\cdot)\) \(\chi_{2003}(182,\cdot)\) \(\chi_{2003}(184,\cdot)\) \(\chi_{2003}(185,\cdot)\) \(\chi_{2003}(190,\cdot)\) \(\chi_{2003}(231,\cdot)\) \(\chi_{2003}(256,\cdot)\) \(\chi_{2003}(298,\cdot)\) \(\chi_{2003}(356,\cdot)\) \(\chi_{2003}(360,\cdot)\) \(\chi_{2003}(390,\cdot)\) \(\chi_{2003}(441,\cdot)\) \(\chi_{2003}(452,\cdot)\) \(\chi_{2003}(477,\cdot)\) \(\chi_{2003}(478,\cdot)\) \(\chi_{2003}(479,\cdot)\) ...

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: $\Q(\zeta_{143})$
Fixed field: Number field defined by a degree 143 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{30}{143}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2003 }(184, a) \) \(1\)\(1\)\(e\left(\frac{70}{143}\right)\)\(e\left(\frac{84}{143}\right)\)\(e\left(\frac{140}{143}\right)\)\(e\left(\frac{30}{143}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{115}{143}\right)\)\(e\left(\frac{67}{143}\right)\)\(e\left(\frac{25}{143}\right)\)\(e\left(\frac{100}{143}\right)\)\(e\left(\frac{40}{143}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2003 }(184,a) \;\) at \(\;a = \) e.g. 2