Properties

Label 1728.1567
Modulus $1728$
Conductor $8$
Order $2$
Real yes
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,1,0]))
 
pari: [g,chi] = znchar(Mod(1567,1728))
 

Basic properties

Modulus: \(1728\)
Conductor: \(8\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{8}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1728.b

\(\chi_{1728}(1567,\cdot)\)

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: \(\Q(\sqrt{-2}) \)

Values on generators

\((703,325,1217)\) → \((-1,-1,1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1728 }(1567, a) \) \(-1\)\(1\)\(-1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1728 }(1567,a) \;\) at \(\;a = \) e.g. 2