Properties

Label 1428.155
Modulus $1428$
Conductor $204$
Order $8$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1428, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,4,0,7]))
 
pari: [g,chi] = znchar(Mod(155,1428))
 

Basic properties

Modulus: \(1428\)
Conductor: \(204\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{204}(155,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1428.bp

\(\chi_{1428}(155,\cdot)\) \(\chi_{1428}(491,\cdot)\) \(\chi_{1428}(995,\cdot)\) \(\chi_{1428}(1079,\cdot)\)

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: 8.8.8508782723328.1

Values on generators

\((715,953,409,1261)\) → \((-1,-1,1,e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1428 }(155, a) \) \(1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1428 }(155,a) \;\) at \(\;a = \) e.g. 2