Properties

Label 5184.647
Modulus $5184$
Conductor $96$
Order $8$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(5184, base_ring=CyclotomicField(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,5,4]))
 
pari: [g,chi] = znchar(Mod(647,5184))
 

Basic properties

Modulus: \(5184\)
Conductor: \(96\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{96}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5184.w

\(\chi_{5184}(647,\cdot)\) \(\chi_{5184}(1943,\cdot)\) \(\chi_{5184}(3239,\cdot)\) \(\chi_{5184}(4535,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.173946175488.1

Values on generators

\((2431,325,1217)\) → \((-1,e\left(\frac{5}{8}\right),-1)\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 5184 }(647, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(1\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(i\)\(e\left(\frac{3}{8}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5184 }(647,a) \;\) at \(\;a = \) e.g. 2