Properties

Label 8619.587
Modulus $8619$
Conductor $663$
Order $24$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,2,3]))
 
pari: [g,chi] = znchar(Mod(587,8619))
 

Basic properties

Modulus: \(8619\)
Conductor: \(663\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{663}(587,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8619.cg

\(\chi_{8619}(587,\cdot)\) \(\chi_{8619}(1103,\cdot)\) \(\chi_{8619}(1709,\cdot)\) \(\chi_{8619}(3629,\cdot)\) \(\chi_{8619}(4751,\cdot)\) \(\chi_{8619}(5051,\cdot)\) \(\chi_{8619}(6173,\cdot)\) \(\chi_{8619}(8600,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: Number field defined by a degree 24 polynomial

Values on generators

\((5747,5917,2536)\) → \((-1,e\left(\frac{1}{12}\right),e\left(\frac{1}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(19\)
\( \chi_{ 8619 }(587, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{24}\right)\)\(1\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8619 }(587,a) \;\) at \(\;a = \) e.g. 2