Properties

Label 896.723
Modulus $896$
Conductor $896$
Order $96$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(896, base_ring=CyclotomicField(96))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([48,21,32]))
 
pari: [g,chi] = znchar(Mod(723,896))
 

Basic properties

Modulus: \(896\)
Conductor: \(896\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(96\)
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 896.bs

\(\chi_{896}(11,\cdot)\) \(\chi_{896}(51,\cdot)\) \(\chi_{896}(67,\cdot)\) \(\chi_{896}(107,\cdot)\) \(\chi_{896}(123,\cdot)\) \(\chi_{896}(163,\cdot)\) \(\chi_{896}(179,\cdot)\) \(\chi_{896}(219,\cdot)\) \(\chi_{896}(235,\cdot)\) \(\chi_{896}(275,\cdot)\) \(\chi_{896}(291,\cdot)\) \(\chi_{896}(331,\cdot)\) \(\chi_{896}(347,\cdot)\) \(\chi_{896}(387,\cdot)\) \(\chi_{896}(403,\cdot)\) \(\chi_{896}(443,\cdot)\) \(\chi_{896}(459,\cdot)\) \(\chi_{896}(499,\cdot)\) \(\chi_{896}(515,\cdot)\) \(\chi_{896}(555,\cdot)\) \(\chi_{896}(571,\cdot)\) \(\chi_{896}(611,\cdot)\) \(\chi_{896}(627,\cdot)\) \(\chi_{896}(667,\cdot)\) \(\chi_{896}(683,\cdot)\) \(\chi_{896}(723,\cdot)\) \(\chi_{896}(739,\cdot)\) \(\chi_{896}(779,\cdot)\) \(\chi_{896}(795,\cdot)\) \(\chi_{896}(835,\cdot)\) ...

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: $\Q(\zeta_{96})$
Fixed field: Number field defined by a degree 96 polynomial

Values on generators

\((127,645,129)\) → \((-1,e\left(\frac{7}{32}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\(-1\)\(1\)\(e\left(\frac{47}{96}\right)\)\(e\left(\frac{85}{96}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{41}{96}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{19}{96}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{37}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 896 }(723,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 896 }(723,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 896 }(723,·),\chi_{ 896 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 896 }(723,·)) \;\) at \(\; a,b = \) e.g. 1,2