# Properties

 Conductor 55 Order 20 Real no Primitive yes Minimal yes Parity odd Orbit label 55.k

# 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(55)

sage: chi = H[3]

pari: [g,chi] = znchar(Mod(3,55))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 55 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 20 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 55.k Orbit index = 11

## 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

$$(12,46)$$ → $$(-i,e\left(\frac{4}{5}\right))$$

## Values

 -1 1 2 3 4 6 7 8 9 12 13 14 $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$-i$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{20})$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 55 }(3,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{55}(3,\cdot)) = \sum_{r\in \Z/55\Z} \chi_{55}(3,r) e\left(\frac{2r}{55}\right) = -1.9294061447+-7.1608234114i$$

## Jacobi sum

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

$$J(\chi_{ 55 }(3,·),\chi_{ 55 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{55}(3,\cdot),\chi_{55}(1,\cdot)) = \sum_{r\in \Z/55\Z} \chi_{55}(3,r) \chi_{55}(1,1-r) = 1$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 55 }(3,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{55}(3,·)) = \sum_{r \in \Z/55\Z} \chi_{55}(3,r) e\left(\frac{1 r + 2 r^{-1}}{55}\right) = 9.2257427016+1.4612140944i$$