Properties

Label 2003.817
Modulus $2003$
Conductor $2003$
Order $182$
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(182))
 
M = H._module
 
chi = DirichletCharacter(H, M([163]))
 
pari: [g,chi] = znchar(Mod(817,2003))
 

Basic properties

Modulus: \(2003\)
Conductor: \(2003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(182\)
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.m

\(\chi_{2003}(6,\cdot)\) \(\chi_{2003}(17,\cdot)\) \(\chi_{2003}(69,\cdot)\) \(\chi_{2003}(132,\cdot)\) \(\chi_{2003}(216,\cdot)\) \(\chi_{2003}(233,\cdot)\) \(\chi_{2003}(260,\cdot)\) \(\chi_{2003}(295,\cdot)\) \(\chi_{2003}(331,\cdot)\) \(\chi_{2003}(334,\cdot)\) \(\chi_{2003}(374,\cdot)\) \(\chi_{2003}(376,\cdot)\) \(\chi_{2003}(382,\cdot)\) \(\chi_{2003}(387,\cdot)\) \(\chi_{2003}(388,\cdot)\) \(\chi_{2003}(392,\cdot)\) \(\chi_{2003}(456,\cdot)\) \(\chi_{2003}(481,\cdot)\) \(\chi_{2003}(502,\cdot)\) \(\chi_{2003}(524,\cdot)\) \(\chi_{2003}(534,\cdot)\) \(\chi_{2003}(567,\cdot)\) \(\chi_{2003}(584,\cdot)\) \(\chi_{2003}(603,\cdot)\) \(\chi_{2003}(604,\cdot)\) \(\chi_{2003}(605,\cdot)\) \(\chi_{2003}(612,\cdot)\) \(\chi_{2003}(707,\cdot)\) \(\chi_{2003}(802,\cdot)\) \(\chi_{2003}(817,\cdot)\) ...

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: $\Q(\zeta_{91})$
Fixed field: Number field defined by a degree 182 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{163}{182}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2003 }(817, a) \) \(-1\)\(1\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{80}{91}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{163}{182}\right)\)\(e\left(\frac{55}{182}\right)\)\(e\left(\frac{29}{182}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{69}{91}\right)\)\(e\left(\frac{29}{91}\right)\)\(e\left(\frac{23}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2003 }(817,a) \;\) at \(\;a = \) e.g. 2