Properties

Label 1011.863
Modulus $1011$
Conductor $1011$
Order $4$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1011, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,3]))
 
pari: [g,chi] = znchar(Mod(863,1011))
 

Basic properties

Modulus: \(1011\)
Conductor: \(1011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
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 1011.f

\(\chi_{1011}(485,\cdot)\) \(\chi_{1011}(863,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.344454777.2

Values on generators

\((338,10)\) → \((-1,-i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1011 }(863, a) \) \(-1\)\(1\)\(-1\)\(1\)\(i\)\(-1\)\(-1\)\(-i\)\(-i\)\(1\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1011 }(863,a) \;\) at \(\;a = \) e.g. 2