Properties

Label 1003.117
Modulus $1003$
Conductor $1003$
Order $8$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([3,4]))
 
pari: [g,chi] = znchar(Mod(117,1003))
 

Basic properties

Modulus: \(1003\)
Conductor: \(1003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
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 1003.h

\(\chi_{1003}(117,\cdot)\) \(\chi_{1003}(648,\cdot)\) \(\chi_{1003}(825,\cdot)\) \(\chi_{1003}(943,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.0.4972221833001953.2

Values on generators

\((768,120)\) → \((e\left(\frac{3}{8}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1003 }(117, a) \) \(-1\)\(1\)\(-i\)\(e\left(\frac{3}{8}\right)\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1003 }(117,a) \;\) at \(\;a = \) e.g. 2