Properties

Label 1143.253
Modulus $1143$
Conductor $127$
Order $2$
Real yes
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1143, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(253,1143))
 

Basic properties

Modulus: \(1143\)
Conductor: \(127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{127}(126,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1143.d

\(\chi_{1143}(253,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{-127}) \)

Values on generators

\((128,892)\) → \((1,-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1143 }(253, a) \) \(-1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1143 }(253,a) \;\) at \(\;a = \) e.g. 2