Properties

Label 2856.251
Modulus $2856$
Conductor $2856$
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(2856, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,2,2,2,1]))
 
pari: [g,chi] = znchar(Mod(251,2856))
 

Basic properties

Modulus: \(2856\)
Conductor: \(2856\)
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 2856.bt

\(\chi_{2856}(251,\cdot)\) \(\chi_{2856}(2435,\cdot)\)

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: \(\mathbb{Q}(i)\)
Fixed field: 4.0.138664512.3

Values on generators

\((2143,1429,953,409,2689)\) → \((-1,-1,-1,-1,i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2856 }(251, a) \) \(-1\)\(1\)\(-i\)\(i\)\(1\)\(1\)\(-i\)\(-1\)\(i\)\(i\)\(-i\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2856 }(251,a) \;\) at \(\;a = \) e.g. 2