Properties

Label 8512.2811
Modulus $8512$
Conductor $8512$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8512, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,3,32,24]))
 
pari: [g,chi] = znchar(Mod(2811,8512))
 

Basic properties

Modulus: \(8512\)
Conductor: \(8512\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
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 8512.kb

\(\chi_{8512}(683,\cdot)\) \(\chi_{8512}(835,\cdot)\) \(\chi_{8512}(1747,\cdot)\) \(\chi_{8512}(1899,\cdot)\) \(\chi_{8512}(2811,\cdot)\) \(\chi_{8512}(2963,\cdot)\) \(\chi_{8512}(3875,\cdot)\) \(\chi_{8512}(4027,\cdot)\) \(\chi_{8512}(4939,\cdot)\) \(\chi_{8512}(5091,\cdot)\) \(\chi_{8512}(6003,\cdot)\) \(\chi_{8512}(6155,\cdot)\) \(\chi_{8512}(7067,\cdot)\) \(\chi_{8512}(7219,\cdot)\) \(\chi_{8512}(8131,\cdot)\) \(\chi_{8512}(8283,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((5055,6917,7297,3137)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{2}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 8512 }(2811, a) \) \(1\)\(1\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{19}{48}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{7}{16}\right)\)\(i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8512 }(2811,a) \;\) at \(\;a = \) e.g. 2