Properties

Label 8993.22
Modulus $8993$
Conductor $8993$
Order $368$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8993, base_ring=CyclotomicField(368))
 
M = H._module
 
chi = DirichletCharacter(H, M([115,104]))
 
pari: [g,chi] = znchar(Mod(22,8993))
 

Basic properties

Modulus: \(8993\)
Conductor: \(8993\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(368\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8993.bd

\(\chi_{8993}(22,\cdot)\) \(\chi_{8993}(45,\cdot)\) \(\chi_{8993}(91,\cdot)\) \(\chi_{8993}(114,\cdot)\) \(\chi_{8993}(160,\cdot)\) \(\chi_{8993}(252,\cdot)\) \(\chi_{8993}(275,\cdot)\) \(\chi_{8993}(367,\cdot)\) \(\chi_{8993}(413,\cdot)\) \(\chi_{8993}(436,\cdot)\) \(\chi_{8993}(482,\cdot)\) \(\chi_{8993}(505,\cdot)\) \(\chi_{8993}(551,\cdot)\) \(\chi_{8993}(643,\cdot)\) \(\chi_{8993}(666,\cdot)\) \(\chi_{8993}(758,\cdot)\) \(\chi_{8993}(804,\cdot)\) \(\chi_{8993}(827,\cdot)\) \(\chi_{8993}(873,\cdot)\) \(\chi_{8993}(896,\cdot)\) \(\chi_{8993}(942,\cdot)\) \(\chi_{8993}(1034,\cdot)\) \(\chi_{8993}(1149,\cdot)\) \(\chi_{8993}(1195,\cdot)\) \(\chi_{8993}(1218,\cdot)\) \(\chi_{8993}(1264,\cdot)\) \(\chi_{8993}(1287,\cdot)\) \(\chi_{8993}(1333,\cdot)\) \(\chi_{8993}(1425,\cdot)\) \(\chi_{8993}(1448,\cdot)\) ...

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: $\Q(\zeta_{368})$
Fixed field: Number field defined by a degree 368 polynomial (not computed)

Values on generators

\((530,7940)\) → \((e\left(\frac{5}{16}\right),e\left(\frac{13}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8993 }(22, a) \) \(1\)\(1\)\(e\left(\frac{165}{184}\right)\)\(e\left(\frac{307}{368}\right)\)\(e\left(\frac{73}{92}\right)\)\(e\left(\frac{311}{368}\right)\)\(e\left(\frac{269}{368}\right)\)\(e\left(\frac{329}{368}\right)\)\(e\left(\frac{127}{184}\right)\)\(e\left(\frac{123}{184}\right)\)\(e\left(\frac{273}{368}\right)\)\(e\left(\frac{29}{368}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8993 }(22,a) \;\) at \(\;a = \) e.g. 2