Properties

Label 1143.82
Modulus $1143$
Conductor $127$
Order $63$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1143, base_ring=CyclotomicField(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,26]))
 
pari: [g,chi] = znchar(Mod(82,1143))
 

Basic properties

Modulus: \(1143\)
Conductor: \(127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(63\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{127}(82,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1143.ca

\(\chi_{1143}(82,\cdot)\) \(\chi_{1143}(136,\cdot)\) \(\chi_{1143}(145,\cdot)\) \(\chi_{1143}(163,\cdot)\) \(\chi_{1143}(199,\cdot)\) \(\chi_{1143}(208,\cdot)\) \(\chi_{1143}(271,\cdot)\) \(\chi_{1143}(280,\cdot)\) \(\chi_{1143}(289,\cdot)\) \(\chi_{1143}(298,\cdot)\) \(\chi_{1143}(316,\cdot)\) \(\chi_{1143}(325,\cdot)\) \(\chi_{1143}(352,\cdot)\) \(\chi_{1143}(415,\cdot)\) \(\chi_{1143}(451,\cdot)\) \(\chi_{1143}(460,\cdot)\) \(\chi_{1143}(469,\cdot)\) \(\chi_{1143}(496,\cdot)\) \(\chi_{1143}(505,\cdot)\) \(\chi_{1143}(523,\cdot)\) \(\chi_{1143}(550,\cdot)\) \(\chi_{1143}(568,\cdot)\) \(\chi_{1143}(577,\cdot)\) \(\chi_{1143}(676,\cdot)\) \(\chi_{1143}(739,\cdot)\) \(\chi_{1143}(748,\cdot)\) \(\chi_{1143}(775,\cdot)\) \(\chi_{1143}(793,\cdot)\) \(\chi_{1143}(811,\cdot)\) \(\chi_{1143}(883,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 63 polynomial

Values on generators

\((128,892)\) → \((1,e\left(\frac{13}{63}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1143 }(82, a) \) \(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{46}{63}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{2}{63}\right)\)\(e\left(\frac{25}{63}\right)\)\(e\left(\frac{37}{63}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1143 }(82,a) \;\) at \(\;a = \) e.g. 2