# Properties

 Modulus 311 Conductor 311 Order 31 Real no Primitive yes Minimal yes Parity even Orbit label 311.e

# Related objects

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(311)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([21]))

pari: [g,chi] = znchar(Mod(47,311))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 311 Conductor = 311 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 31 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 = even Orbit label = 311.e Orbit index = 5

## Galois orbit

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$17$$ → $$e\left(\frac{21}{31}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{28}{31}\right)$$ $$e\left(\frac{3}{31}\right)$$ $$e\left(\frac{25}{31}\right)$$ $$e\left(\frac{24}{31}\right)$$ $$1$$ $$e\left(\frac{26}{31}\right)$$ $$e\left(\frac{22}{31}\right)$$ $$e\left(\frac{6}{31}\right)$$ $$e\left(\frac{21}{31}\right)$$ $$e\left(\frac{14}{31}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{31})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 311 }(47,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{311}(47,\cdot)) = \sum_{r\in \Z/311\Z} \chi_{311}(47,r) e\left(\frac{2r}{311}\right) = 14.3425184072+-10.261197091i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 311 }(47,·),\chi_{ 311 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{311}(47,\cdot),\chi_{311}(1,\cdot)) = \sum_{r\in \Z/311\Z} \chi_{311}(47,r) \chi_{311}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 311 }(47,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{311}(47,·)) = \sum_{r \in \Z/311\Z} \chi_{311}(47,r) e\left(\frac{1 r + 2 r^{-1}}{311}\right) = -13.9110929261+4.3646335631i$$