Properties

Label 1680.139
Modulus $1680$
Conductor $560$
Order $4$
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(1680, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,1,0,2,2]))
 
pari: [g,chi] = znchar(Mod(139,1680))
 

Basic properties

Modulus: \(1680\)
Conductor: \(560\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{560}(139,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1680.ch

\(\chi_{1680}(139,\cdot)\) \(\chi_{1680}(979,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.4.2508800.1

Values on generators

\((1471,421,1121,337,241)\) → \((-1,i,1,-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1680 }(139, a) \) \(1\)\(1\)\(-i\)\(-i\)\(1\)\(-i\)\(-1\)\(-i\)\(1\)\(-i\)\(1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1680 }(139,a) \;\) at \(\;a = \) e.g. 2