Properties

Label 1856.505
Modulus $1856$
Conductor $928$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,5,2]))
 
pari: [g,chi] = znchar(Mod(505,1856))
 

Basic properties

Modulus: \(1856\)
Conductor: \(928\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{928}(853,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1856.z

\(\chi_{1856}(505,\cdot)\) \(\chi_{1856}(713,\cdot)\) \(\chi_{1856}(1433,\cdot)\) \(\chi_{1856}(1641,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.0.1277373355296555008.36

Values on generators

\((639,581,321)\) → \((1,e\left(\frac{5}{8}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1856 }(505, a) \) \(-1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(i\)\(i\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1856 }(505,a) \;\) at \(\;a = \) e.g. 2