Properties

Label 6815.2727
Modulus $6815$
Conductor $5$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6815, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,0,0]))
 
pari: [g,chi] = znchar(Mod(2727,6815))
 

Basic properties

Modulus: \(6815\)
Conductor: \(5\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{5}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6815.q

\(\chi_{6815}(2727,\cdot)\) \(\chi_{6815}(5453,\cdot)\)

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: \(\Q(\zeta_{5})\)

Values on generators

\((2727,2351,146)\) → \((i,1,1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 6815 }(2727, a) \) \(-1\)\(1\)\(i\)\(-i\)\(-1\)\(1\)\(i\)\(-i\)\(-1\)\(1\)\(i\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6815 }(2727,a) \;\) at \(\;a = \) e.g. 2