Properties

Label 896.269
Modulus $896$
Conductor $896$
Order $96$
Real no
Primitive yes
Minimal yes
Parity odd

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([0,45,16]))
 
pari: [g,chi] = znchar(Mod(269,896))
 

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.bv

\(\chi_{896}(5,\cdot)\) \(\chi_{896}(45,\cdot)\) \(\chi_{896}(61,\cdot)\) \(\chi_{896}(101,\cdot)\) \(\chi_{896}(117,\cdot)\) \(\chi_{896}(157,\cdot)\) \(\chi_{896}(173,\cdot)\) \(\chi_{896}(213,\cdot)\) \(\chi_{896}(229,\cdot)\) \(\chi_{896}(269,\cdot)\) \(\chi_{896}(285,\cdot)\) \(\chi_{896}(325,\cdot)\) \(\chi_{896}(341,\cdot)\) \(\chi_{896}(381,\cdot)\) \(\chi_{896}(397,\cdot)\) \(\chi_{896}(437,\cdot)\) \(\chi_{896}(453,\cdot)\) \(\chi_{896}(493,\cdot)\) \(\chi_{896}(509,\cdot)\) \(\chi_{896}(549,\cdot)\) \(\chi_{896}(565,\cdot)\) \(\chi_{896}(605,\cdot)\) \(\chi_{896}(621,\cdot)\) \(\chi_{896}(661,\cdot)\) \(\chi_{896}(677,\cdot)\) \(\chi_{896}(717,\cdot)\) \(\chi_{896}(733,\cdot)\) \(\chi_{896}(773,\cdot)\) \(\chi_{896}(789,\cdot)\) \(\chi_{896}(829,\cdot)\) ...

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

\((127,645,129)\) → \((1,e\left(\frac{15}{32}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 896 }(269, a) \) \(-1\)\(1\)\(e\left(\frac{55}{96}\right)\)\(e\left(\frac{29}{96}\right)\)\(e\left(\frac{7}{48}\right)\)\(e\left(\frac{49}{96}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{59}{96}\right)\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{29}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 896 }(269,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 896 }(269,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 896 }(269,·),\chi_{ 896 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 896 }(269,·)) \;\) at \(\; a,b = \) e.g. 1,2