Properties

Conductor 68
Order 4
Real No
Primitive Yes
Parity Odd
Orbit Label 68.f

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(68)
sage: chi = H[55]
pari: [g,chi] = znchar(Mod(55,68))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 68
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 = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 68.f
Orbit index = 6

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 ]

\(\chi_{68}(47,\cdot)\) \(\chi_{68}(55,\cdot)\)

Values on generators

\((35,37)\) → \((-1,-i)\)

Values

-113579111315192123
\(-1\)\(1\)\(i\)\(-i\)\(-i\)\(-1\)\(-i\)\(1\)\(1\)\(1\)\(1\)\(-i\)
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)
\( \tau_{ a }( \chi_{ 68 }(55,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{68}(55,\cdot)) = \sum_{r\in \Z/68\Z} \chi_{68}(55,r) e\left(\frac{r}{34}\right) = -0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 68 }(55,·),\chi_{ 68 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{68}(55,\cdot),\chi_{68}(1,\cdot)) = \sum_{r\in \Z/68\Z} \chi_{68}(55,r) \chi_{68}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 68 }(55,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{68}(55,·)) = \sum_{r \in \Z/68\Z} \chi_{68}(55,r) e\left(\frac{1 r + 2 r^{-1}}{68}\right) = 13.6467382206 \)