Properties

Label 8512.7817
Modulus $8512$
Conductor $4256$
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(8512, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,20,4]))
 
pari: [g,chi] = znchar(Mod(7817,8512))
 

Basic properties

Modulus: \(8512\)
Conductor: \(4256\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4256}(4093,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8512.hq

\(\chi_{8512}(297,\cdot)\) \(\chi_{8512}(1433,\cdot)\) \(\chi_{8512}(2425,\cdot)\) \(\chi_{8512}(3561,\cdot)\) \(\chi_{8512}(4553,\cdot)\) \(\chi_{8512}(5689,\cdot)\) \(\chi_{8512}(6681,\cdot)\) \(\chi_{8512}(7817,\cdot)\)

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

\((5055,6917,7297,3137)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{5}{6}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 8512 }(7817, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{5}{24}\right)\)\(i\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8512 }(7817,a) \;\) at \(\;a = \) e.g. 2