Properties

Label 3584.11
Modulus $3584$
Conductor $3584$
Order $384$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3584, base_ring=CyclotomicField(384))
 
M = H._module
 
chi = DirichletCharacter(H, M([192,255,256]))
 
pari: [g,chi] = znchar(Mod(11,3584))
 

Basic properties

Modulus: \(3584\)
Conductor: \(3584\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(384\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3584.cj

\(\chi_{3584}(11,\cdot)\) \(\chi_{3584}(51,\cdot)\) \(\chi_{3584}(67,\cdot)\) \(\chi_{3584}(107,\cdot)\) \(\chi_{3584}(123,\cdot)\) \(\chi_{3584}(163,\cdot)\) \(\chi_{3584}(179,\cdot)\) \(\chi_{3584}(219,\cdot)\) \(\chi_{3584}(235,\cdot)\) \(\chi_{3584}(275,\cdot)\) \(\chi_{3584}(291,\cdot)\) \(\chi_{3584}(331,\cdot)\) \(\chi_{3584}(347,\cdot)\) \(\chi_{3584}(387,\cdot)\) \(\chi_{3584}(403,\cdot)\) \(\chi_{3584}(443,\cdot)\) \(\chi_{3584}(459,\cdot)\) \(\chi_{3584}(499,\cdot)\) \(\chi_{3584}(515,\cdot)\) \(\chi_{3584}(555,\cdot)\) \(\chi_{3584}(571,\cdot)\) \(\chi_{3584}(611,\cdot)\) \(\chi_{3584}(627,\cdot)\) \(\chi_{3584}(667,\cdot)\) \(\chi_{3584}(683,\cdot)\) \(\chi_{3584}(723,\cdot)\) \(\chi_{3584}(739,\cdot)\) \(\chi_{3584}(779,\cdot)\) \(\chi_{3584}(795,\cdot)\) \(\chi_{3584}(835,\cdot)\) ...

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: $\Q(\zeta_{384})$
Fixed field: Number field defined by a degree 384 polynomial (not computed)

Values on generators

\((1023,1541,1025)\) → \((-1,e\left(\frac{85}{128}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 3584 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{157}{384}\right)\)\(e\left(\frac{383}{384}\right)\)\(e\left(\frac{157}{192}\right)\)\(e\left(\frac{235}{384}\right)\)\(e\left(\frac{91}{128}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{25}{96}\right)\)\(e\left(\frac{41}{384}\right)\)\(e\left(\frac{25}{192}\right)\)\(e\left(\frac{191}{192}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3584 }(11,a) \;\) at \(\;a = \) e.g. 2