Properties

Label 3584.65
Modulus $3584$
Conductor $224$
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(3584, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,8]))
 
pari: [g,chi] = znchar(Mod(65,3584))
 

Basic properties

Modulus: \(3584\)
Conductor: \(224\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{224}(205,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3584.bh

\(\chi_{3584}(65,\cdot)\) \(\chi_{3584}(193,\cdot)\) \(\chi_{3584}(961,\cdot)\) \(\chi_{3584}(1089,\cdot)\) \(\chi_{3584}(1857,\cdot)\) \(\chi_{3584}(1985,\cdot)\) \(\chi_{3584}(2753,\cdot)\) \(\chi_{3584}(2881,\cdot)\)

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: 24.24.329123002999201416128761938882499016916992.1

Values on generators

\((1023,1541,1025)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 3584 }(65, a) \) \(1\)\(1\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3584 }(65,a) \;\) at \(\;a = \) e.g. 2