Properties

Label 1017.98
Modulus $1017$
Conductor $339$
Order $4$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1017, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,1]))
 
pari: [g,chi] = znchar(Mod(98,1017))
 

Basic properties

Modulus: \(1017\)
Conductor: \(339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{339}(98,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1017.f

\(\chi_{1017}(98,\cdot)\) \(\chi_{1017}(467,\cdot)\)

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: 4.0.12986073.1

Values on generators

\((227,568)\) → \((-1,i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1017 }(98, a) \) \(-1\)\(1\)\(-1\)\(1\)\(i\)\(1\)\(-1\)\(-i\)\(1\)\(-1\)\(-1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1017 }(98,a) \;\) at \(\;a = \) e.g. 2