# Properties

 Conductor 731 Order 84 Real No Primitive Yes Parity Even Orbit Label 731.bh

# 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(731)

sage: chi = H[13]

pari: [g,chi] = znchar(Mod(13,731))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 731 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 84 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 = 731.bh Orbit index = 34

## 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

$$(173,562)$$ → $$(i,e\left(\frac{16}{21}\right))$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{17}{28}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{84})$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 731 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{731}(13,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(13,r) e\left(\frac{2r}{731}\right) = -7.4372345375+25.9939905062i$$

## Jacobi sum

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

$$J(\chi_{ 731 }(13,·),\chi_{ 731 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{731}(13,\cdot),\chi_{731}(1,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(13,r) \chi_{731}(1,1-r) = 1$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 731 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{731}(13,·)) = \sum_{r \in \Z/731\Z} \chi_{731}(13,r) e\left(\frac{1 r + 2 r^{-1}}{731}\right) = -0.5836813961+-0.1332214698i$$