Properties

Label 1560.203
Modulus $1560$
Conductor $1560$
Order $4$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,2,2,3,1]))
 
pari: [g,chi] = znchar(Mod(203,1560))
 

Basic properties

Modulus: \(1560\)
Conductor: \(1560\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1560.cx

\(\chi_{1560}(203,\cdot)\) \(\chi_{1560}(707,\cdot)\)

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: 4.4.158184000.2

Values on generators

\((391,781,521,937,1081)\) → \((-1,-1,-1,-i,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1560 }(203, a) \) \(1\)\(1\)\(1\)\(i\)\(-i\)\(-i\)\(-i\)\(-1\)\(-i\)\(1\)\(-i\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1560 }(203,a) \;\) at \(\;a = \) e.g. 2