# Properties

 Conductor 71 Order 35 Real No Primitive Yes Parity Even Orbit Label 71.g

# Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(71)

sage: chi = H

pari: [g,chi] = znchar(Mod(15,71))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 71 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 35 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 = Even Orbit label = 71.g Orbit index = 7

## 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 ]

## Values on generators

$$7$$ → $$e\left(\frac{27}{35}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{22}{35}\right)$$ $$e\left(\frac{2}{35}\right)$$ $$e\left(\frac{9}{35}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{24}{35}\right)$$ $$e\left(\frac{27}{35}\right)$$ $$e\left(\frac{31}{35}\right)$$ $$e\left(\frac{4}{35}\right)$$ $$e\left(\frac{8}{35}\right)$$ $$e\left(\frac{32}{35}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{35})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 71 }(15,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{71}(15,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(15,r) e\left(\frac{2r}{71}\right) = 1.7408380343+8.2443606749i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)

$$J(\chi_{ 71 }(15,·),\chi_{ 71 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{71}(15,\cdot),\chi_{71}(1,\cdot)) = \sum_{r\in \Z/71\Z} \chi_{71}(15,r) \chi_{71}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)

$$K(a,b,\chi_{ 71 }(15,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{71}(15,·)) = \sum_{r \in \Z/71\Z} \chi_{71}(15,r) e\left(\frac{1 r + 2 r^{-1}}{71}\right) = -3.901028704+9.1269103643i$$