Properties

Conductor 35
Order 6
Real No
Primitive No
Parity Odd
Orbit Label 70.h

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(70)
sage: chi = H[19]
pari: [g,chi] = znchar(Mod(19,70))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 35
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 6
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 = 70.h
Orbit index = 8

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 ]

\(\chi_{70}(19,\cdot)\) \(\chi_{70}(59,\cdot)\)

Inducing primitive character

\(\chi_{35}(19,\cdot)\)

Values on generators

\((57,31)\) → \((-1,e\left(\frac{5}{6}\right))\)

Values

-11391113171923272931
\(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(1\)\(e\left(\frac{5}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{3})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 70 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{70}(19,\cdot)) = \sum_{r\in \Z/70\Z} \chi_{70}(19,r) e\left(\frac{r}{35}\right) = -0.7478592565+-5.8686204965i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 70 }(19,·),\chi_{ 70 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{70}(19,\cdot),\chi_{70}(1,\cdot)) = \sum_{r\in \Z/70\Z} \chi_{70}(19,r) \chi_{70}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 70 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{70}(19,·)) = \sum_{r \in \Z/70\Z} \chi_{70}(19,r) e\left(\frac{1 r + 2 r^{-1}}{70}\right) = 0.0 \)