Properties

Label 8993.18
Modulus $8993$
Conductor $529$
Order $253$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8993, base_ring=CyclotomicField(506))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,232]))
 
pari: [g,chi] = znchar(Mod(18,8993))
 

Basic properties

Modulus: \(8993\)
Conductor: \(529\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(253\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{529}(18,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8993.bc

\(\chi_{8993}(18,\cdot)\) \(\chi_{8993}(35,\cdot)\) \(\chi_{8993}(52,\cdot)\) \(\chi_{8993}(154,\cdot)\) \(\chi_{8993}(188,\cdot)\) \(\chi_{8993}(239,\cdot)\) \(\chi_{8993}(256,\cdot)\) \(\chi_{8993}(307,\cdot)\) \(\chi_{8993}(324,\cdot)\) \(\chi_{8993}(358,\cdot)\) \(\chi_{8993}(409,\cdot)\) \(\chi_{8993}(426,\cdot)\) \(\chi_{8993}(443,\cdot)\) \(\chi_{8993}(545,\cdot)\) \(\chi_{8993}(579,\cdot)\) \(\chi_{8993}(630,\cdot)\) \(\chi_{8993}(698,\cdot)\) \(\chi_{8993}(715,\cdot)\) \(\chi_{8993}(749,\cdot)\) \(\chi_{8993}(800,\cdot)\) \(\chi_{8993}(817,\cdot)\) \(\chi_{8993}(834,\cdot)\) \(\chi_{8993}(936,\cdot)\) \(\chi_{8993}(970,\cdot)\) \(\chi_{8993}(1021,\cdot)\) \(\chi_{8993}(1038,\cdot)\) \(\chi_{8993}(1089,\cdot)\) \(\chi_{8993}(1106,\cdot)\) \(\chi_{8993}(1140,\cdot)\) \(\chi_{8993}(1191,\cdot)\) ...

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: $\Q(\zeta_{253})$
Fixed field: Number field defined by a degree 253 polynomial (not computed)

Values on generators

\((530,7940)\) → \((1,e\left(\frac{116}{253}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8993 }(18, a) \) \(1\)\(1\)\(e\left(\frac{177}{253}\right)\)\(e\left(\frac{85}{253}\right)\)\(e\left(\frac{101}{253}\right)\)\(e\left(\frac{116}{253}\right)\)\(e\left(\frac{9}{253}\right)\)\(e\left(\frac{37}{253}\right)\)\(e\left(\frac{25}{253}\right)\)\(e\left(\frac{170}{253}\right)\)\(e\left(\frac{40}{253}\right)\)\(e\left(\frac{219}{253}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8993 }(18,a) \;\) at \(\;a = \) e.g. 2