# Properties

 Conductor 31 Order 15 Real No Primitive No Parity Even Orbit Label 403.bi

# 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(403)

sage: chi = H[14]

pari: [g,chi] = znchar(Mod(14,403))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 31 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 15 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 = Even Orbit label = 403.bi Orbit index = 35

## 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

$$(249,313)$$ → $$(1,e\left(\frac{11}{15}\right))$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{15})$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 403 }(14,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{403}(14,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(14,r) e\left(\frac{2r}{403}\right) = -4.2138648197+-3.6391404592i$$

## Jacobi sum

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

$$J(\chi_{ 403 }(14,·),\chi_{ 403 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{403}(14,\cdot),\chi_{403}(1,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(14,r) \chi_{403}(1,1-r) = -11$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 403 }(14,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{403}(14,·)) = \sum_{r \in \Z/403\Z} \chi_{403}(14,r) e\left(\frac{1 r + 2 r^{-1}}{403}\right) = -0.1513405919+0.4657784482i$$