Properties

Conductor 136
Order 2
Real Yes
Primitive Yes
Parity Odd
Orbit Label 136.e

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(136)
sage: chi = H[67]
pari: [g,chi] = znchar(Mod(67,136))

Kronecker symbol representation

sage: kronecker_character(-136)
pari: znchartokronecker(g,chi)

\(\displaystyle\left(\frac{-136}{\bullet}\right)\)

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 136
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 2
Real = Yes
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 = 136.e
Orbit index = 5

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 ]

\(\chi_{136}(67,\cdot)\)

Values on generators

\((103,69,105)\) → \((-1,-1,-1)\)

Values

-113579111315192123
\(-1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)\(-1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 136 }(67,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{136}(67,\cdot)) = \sum_{r\in \Z/136\Z} \chi_{136}(67,r) e\left(\frac{r}{68}\right) = 0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 136 }(67,·),\chi_{ 136 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{136}(67,\cdot),\chi_{136}(1,\cdot)) = \sum_{r\in \Z/136\Z} \chi_{136}(67,r) \chi_{136}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 136 }(67,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{136}(67,·)) = \sum_{r \in \Z/136\Z} \chi_{136}(67,r) e\left(\frac{1 r + 2 r^{-1}}{136}\right) = -19.8303008745i \)