Properties

Label 1083.457
Modulus $1083$
Conductor $361$
Order $19$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1083, base_ring=CyclotomicField(38))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,16]))
 
pari: [g,chi] = znchar(Mod(457,1083))
 

Basic properties

Modulus: \(1083\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(19\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{361}(96,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1083.m

\(\chi_{1083}(58,\cdot)\) \(\chi_{1083}(115,\cdot)\) \(\chi_{1083}(172,\cdot)\) \(\chi_{1083}(229,\cdot)\) \(\chi_{1083}(286,\cdot)\) \(\chi_{1083}(343,\cdot)\) \(\chi_{1083}(400,\cdot)\) \(\chi_{1083}(457,\cdot)\) \(\chi_{1083}(514,\cdot)\) \(\chi_{1083}(571,\cdot)\) \(\chi_{1083}(628,\cdot)\) \(\chi_{1083}(685,\cdot)\) \(\chi_{1083}(742,\cdot)\) \(\chi_{1083}(799,\cdot)\) \(\chi_{1083}(856,\cdot)\) \(\chi_{1083}(913,\cdot)\) \(\chi_{1083}(970,\cdot)\) \(\chi_{1083}(1027,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 19.19.10842505080063916320800450434338728415281531281.1

Values on generators

\((362,724)\) → \((1,e\left(\frac{8}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1083 }(457, a) \) \(1\)\(1\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{10}{19}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{13}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1083 }(457,a) \;\) at \(\;a = \) e.g. 2