Properties

Label 4235.3389
Modulus $4235$
Conductor $5$
Order $2$
Real yes
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,0,0]))
 
pari: [g,chi] = znchar(Mod(3389,4235))
 

Basic properties

Modulus: \(4235\)
Conductor: \(5\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{5}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4235.b

\(\chi_{4235}(3389,\cdot)\)

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: \(\Q(\sqrt{5}) \)

Values on generators

\((2542,1816,2906)\) → \((-1,1,1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 4235 }(3389, a) \) \(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4235 }(3389,a) \;\) at \(\;a = \) e.g. 2