Properties

Label 255.19
Modulus $255$
Conductor $85$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(255, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,4,7]))
 
pari: [g,chi] = znchar(Mod(19,255))
 

Basic properties

Modulus: \(255\)
Conductor: \(85\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{85}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 255.z

\(\chi_{255}(19,\cdot)\) \(\chi_{255}(49,\cdot)\) \(\chi_{255}(94,\cdot)\) \(\chi_{255}(229,\cdot)\)

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: 8.8.256461670625.1

Values on generators

\((86,52,241)\) → \((1,-1,e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(19\)\(22\)
\( \chi_{ 255 }(19, a) \) \(1\)\(1\)\(-i\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(i\)\(e\left(\frac{1}{8}\right)\)\(1\)\(e\left(\frac{7}{8}\right)\)\(1\)\(i\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 255 }(19,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 255 }(19,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 255 }(19,·),\chi_{ 255 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 255 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2