Properties

Label 3584.1441
Modulus $3584$
Conductor $448$
Order $16$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3584, base_ring=CyclotomicField(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,3,8]))
 
pari: [g,chi] = znchar(Mod(1441,3584))
 

Basic properties

Modulus: \(3584\)
Conductor: \(448\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{448}(125,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3584.bf

\(\chi_{3584}(97,\cdot)\) \(\chi_{3584}(545,\cdot)\) \(\chi_{3584}(993,\cdot)\) \(\chi_{3584}(1441,\cdot)\) \(\chi_{3584}(1889,\cdot)\) \(\chi_{3584}(2337,\cdot)\) \(\chi_{3584}(2785,\cdot)\) \(\chi_{3584}(3233,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{16})\)
Fixed field: 16.0.3484608386920116940487669055488.4

Values on generators

\((1023,1541,1025)\) → \((1,e\left(\frac{3}{16}\right),-1)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 3584 }(1441, a) \) \(-1\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(-i\)\(-i\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3584 }(1441,a) \;\) at \(\;a = \) e.g. 2