Properties

Label 6840.1673
Modulus $6840$
Conductor $15$
Order $4$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6840, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,2,3,0]))
 
pari: [g,chi] = znchar(Mod(1673,6840))
 

Basic properties

Modulus: \(6840\)
Conductor: \(15\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{15}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6840.bp

\(\chi_{6840}(1673,\cdot)\) \(\chi_{6840}(5777,\cdot)\)

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: \(\Q(\zeta_{15})^+\)

Values on generators

\((1711,3421,5321,2737,6481)\) → \((1,1,-1,-i,1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 6840 }(1673, a) \) \(1\)\(1\)\(-i\)\(-1\)\(i\)\(i\)\(-i\)\(1\)\(1\)\(-i\)\(-1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6840 }(1673,a) \;\) at \(\;a = \) e.g. 2