# Properties

 Modulus 77 Conductor 77 Order 10 Real no Primitive yes Minimal yes Parity even Orbit label 77.l

# Learn more about

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

sage: H = DirichletGroup(77)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([5,1]))

pari: [g,chi] = znchar(Mod(13,77))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 77 Conductor = 77 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 10 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 = 77.l Orbit index = 12

## 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

$$(45,57)$$ → $$(-1,e\left(\frac{1}{10}\right))$$

## Values

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

## Related number fields

 Field of values $$\Q(\zeta_{5})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 77 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{77}(13,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(13,r) e\left(\frac{2r}{77}\right) = 5.3126141483+6.9839910446i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 77 }(13,·),\chi_{ 77 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{77}(13,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(13,r) \chi_{77}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 77 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{77}(13,·)) = \sum_{r \in \Z/77\Z} \chi_{77}(13,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right) = 6.4812517375+2.1058863458i$$