Properties

Label 1037.489
Modulus $1037$
Conductor $17$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1037, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,0]))
 
pari: [g,chi] = znchar(Mod(489,1037))
 

Basic properties

Modulus: \(1037\)
Conductor: \(17\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{17}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1037.f

\(\chi_{1037}(123,\cdot)\) \(\chi_{1037}(489,\cdot)\)

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: \(\mathbb{Q}(i)\)
Fixed field: 4.4.4913.1

Values on generators

\((428,307)\) → \((i,1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1037 }(489, a) \) \(1\)\(1\)\(-1\)\(i\)\(1\)\(i\)\(-i\)\(-i\)\(-1\)\(-1\)\(-i\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1037 }(489,a) \;\) at \(\;a = \) e.g. 2