Properties

Label 4012.3775
Modulus $4012$
Conductor $236$
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(4012, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,0,1]))
 
pari: [g,chi] = znchar(Mod(3775,4012))
 

Basic properties

Modulus: \(4012\)
Conductor: \(236\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{236}(235,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4012.e

\(\chi_{4012}(3775,\cdot)\)

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: \(\Q(\sqrt{59}) \)

Values on generators

\((2007,3777,3129)\) → \((-1,1,-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 4012 }(3775, a) \) \(1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4012 }(3775,a) \;\) at \(\;a = \) e.g. 2