Properties

Conductor 48
Order 4
Real No
Primitive No
Parity Odd
Orbit Label 96.i

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(96)
sage: chi = H[89]
pari: [g,chi] = znchar(Mod(89,96))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 48
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 4
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 96.i
Orbit index = 9

Galois orbit

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

\(\chi_{96}(41,\cdot)\) \(\chi_{96}(89,\cdot)\)

Inducing primitive character

\(\chi_{48}(5,\cdot)\)

Values on generators

\((31,37,65)\) → \((1,i,-1)\)

Values

-11571113171923252931
\(-1\)\(1\)\(-i\)\(-1\)\(-i\)\(-i\)\(-1\)\(-i\)\(1\)\(-1\)\(i\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(i)\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 96 }(89,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{96}(89,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(89,r) e\left(\frac{r}{48}\right) = 12.8016503231+-5.3026171846i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 96 }(89,·),\chi_{ 96 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{96}(89,\cdot),\chi_{96}(1,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(89,r) \chi_{96}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 96 }(89,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{96}(89,·)) = \sum_{r \in \Z/96\Z} \chi_{96}(89,r) e\left(\frac{1 r + 2 r^{-1}}{96}\right) = -0.0 \)