Properties

Label 896.131
Modulus $896$
Conductor $896$
Order $96$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(96))
 
M = H._module
 
chi = DirichletCharacter(H, M([48,9,80]))
 
pari: [g,chi] = znchar(Mod(131,896))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 896.bt

\(\chi_{896}(3,\cdot)\) \(\chi_{896}(19,\cdot)\) \(\chi_{896}(59,\cdot)\) \(\chi_{896}(75,\cdot)\) \(\chi_{896}(115,\cdot)\) \(\chi_{896}(131,\cdot)\) \(\chi_{896}(171,\cdot)\) \(\chi_{896}(187,\cdot)\) \(\chi_{896}(227,\cdot)\) \(\chi_{896}(243,\cdot)\) \(\chi_{896}(283,\cdot)\) \(\chi_{896}(299,\cdot)\) \(\chi_{896}(339,\cdot)\) \(\chi_{896}(355,\cdot)\) \(\chi_{896}(395,\cdot)\) \(\chi_{896}(411,\cdot)\) \(\chi_{896}(451,\cdot)\) \(\chi_{896}(467,\cdot)\) \(\chi_{896}(507,\cdot)\) \(\chi_{896}(523,\cdot)\) \(\chi_{896}(563,\cdot)\) \(\chi_{896}(579,\cdot)\) \(\chi_{896}(619,\cdot)\) \(\chi_{896}(635,\cdot)\) \(\chi_{896}(675,\cdot)\) \(\chi_{896}(691,\cdot)\) \(\chi_{896}(731,\cdot)\) \(\chi_{896}(747,\cdot)\) \(\chi_{896}(787,\cdot)\) \(\chi_{896}(803,\cdot)\) ...

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

Values on generators

\((127,645,129)\) → \((-1,e\left(\frac{3}{32}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 896 }(131, a) \) \(1\)\(1\)\(e\left(\frac{59}{96}\right)\)\(e\left(\frac{25}{96}\right)\)\(e\left(\frac{11}{48}\right)\)\(e\left(\frac{77}{96}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{79}{96}\right)\)\(e\left(\frac{23}{48}\right)\)\(e\left(\frac{25}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 896 }(131,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 896 }(131,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 896 }(131,·),\chi_{ 896 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 896 }(131,·)) \;\) at \(\; a,b = \) e.g. 1,2