# Properties

 Modulus 79 Conductor 79 Order 13 Real no Primitive yes Minimal yes Parity even Orbit label 79.e

# Learn more about

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

sage: H = DirichletGroup(79)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([12]))

pari: [g,chi] = znchar(Mod(22,79))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 79 Conductor = 79 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 13 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 = 79.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

$$3$$ → $$e\left(\frac{12}{13}\right)$$

## Values

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

## Related number fields

 Field of values $$\Q(\zeta_{13})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 79 }(22,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{79}(22,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(22,r) e\left(\frac{2r}{79}\right) = 6.3221917382+6.2473907854i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 79 }(22,·),\chi_{ 79 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{79}(22,\cdot),\chi_{79}(1,\cdot)) = \sum_{r\in \Z/79\Z} \chi_{79}(22,r) \chi_{79}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 79 }(22,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{79}(22,·)) = \sum_{r \in \Z/79\Z} \chi_{79}(22,r) e\left(\frac{1 r + 2 r^{-1}}{79}\right) = 4.3764997607+-6.3404545219i$$