Properties

Label 2003.364
Modulus $2003$
Conductor $2003$
Order $286$
Real no
Primitive yes
Minimal yes
Parity odd

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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([41]))
 
pari: [g,chi] = znchar(Mod(364,2003))
 

Basic properties

Modulus: \(2003\)
Conductor: \(2003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(286\)
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 2003.n

\(\chi_{2003}(2,\cdot)\) \(\chi_{2003}(8,\cdot)\) \(\chi_{2003}(11,\cdot)\) \(\chi_{2003}(21,\cdot)\) \(\chi_{2003}(23,\cdot)\) \(\chi_{2003}(32,\cdot)\) \(\chi_{2003}(44,\cdot)\) \(\chi_{2003}(67,\cdot)\) \(\chi_{2003}(71,\cdot)\) \(\chi_{2003}(84,\cdot)\) \(\chi_{2003}(92,\cdot)\) \(\chi_{2003}(128,\cdot)\) \(\chi_{2003}(149,\cdot)\) \(\chi_{2003}(176,\cdot)\) \(\chi_{2003}(178,\cdot)\) \(\chi_{2003}(195,\cdot)\) \(\chi_{2003}(226,\cdot)\) \(\chi_{2003}(239,\cdot)\) \(\chi_{2003}(242,\cdot)\) \(\chi_{2003}(249,\cdot)\) \(\chi_{2003}(268,\cdot)\) \(\chi_{2003}(284,\cdot)\) \(\chi_{2003}(310,\cdot)\) \(\chi_{2003}(313,\cdot)\) \(\chi_{2003}(336,\cdot)\) \(\chi_{2003}(348,\cdot)\) \(\chi_{2003}(364,\cdot)\) \(\chi_{2003}(368,\cdot)\) \(\chi_{2003}(370,\cdot)\) \(\chi_{2003}(377,\cdot)\) ...

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 286 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{41}{286}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2003 }(364, a) \) \(-1\)\(1\)\(e\left(\frac{191}{286}\right)\)\(e\left(\frac{86}{143}\right)\)\(e\left(\frac{48}{143}\right)\)\(e\left(\frac{41}{286}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{181}{286}\right)\)\(e\left(\frac{1}{286}\right)\)\(e\left(\frac{29}{143}\right)\)\(e\left(\frac{116}{143}\right)\)\(e\left(\frac{7}{286}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2003 }(364,a) \;\) at \(\;a = \) e.g. 2