# Properties

 Conductor 103 Order 34 Real No Primitive Yes Parity Odd Orbit Label 103.f

# 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(103)
sage: chi = H[3]
pari: [g,chi] = znchar(Mod(3,103))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 103 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 34 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 = 103.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 ]

## Values on generators

$$5$$ → $$e\left(\frac{13}{34}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{31}{34}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{13}{34}\right)$$ $$e\left(\frac{25}{34}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{7}{34}\right)$$ $$e\left(\frac{11}{34}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{17})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 103 }(3,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{103}(3,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(3,r) e\left(\frac{2r}{103}\right) = 3.0014580474+-9.6949084364i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 103 }(3,·),\chi_{ 103 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{103}(3,\cdot),\chi_{103}(1,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(3,r) \chi_{103}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 103 }(3,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{103}(3,·)) = \sum_{r \in \Z/103\Z} \chi_{103}(3,r) e\left(\frac{1 r + 2 r^{-1}}{103}\right) = -0.745836584+-1.2045674426i$$