↑

Properties

Label 96.91
Modulus $96$
Conductor $32$
Order $8$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(96, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,1,0]))
 
Copy content pari:[g,chi] = znchar(Mod(91,96))
 

Basic properties

Modulus: \(96\)
Conductor: \(32\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{32}(27,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 96.m

\(\chi_{96}(19,\cdot)\) \(\chi_{96}(43,\cdot)\) \(\chi_{96}(67,\cdot)\) \(\chi_{96}(91,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.0.2147483648.1

Values on generators

\((31,37,65)\) → \((-1,e\left(\frac{1}{8}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 96 }(91, a) \) \(-1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(i\)\(i\)\(e\left(\frac{3}{8}\right)\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 96 }(91,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 96 }(91,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 96 }(91,·),\chi_{ 96 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 96 }(91,·)) \;\) at \(\; a,b = \) e.g. 1,2