# Properties

 Conductor 77 Order 30 Real no Primitive yes Minimal yes Parity odd Orbit label 77.o

# 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(77)

sage: chi = H[2]

pari: [g,chi] = znchar(Mod(2,77))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 77 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 30 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 = 77.o Orbit index = 15

## 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

$$(45,57)$$ → $$(e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right))$$

## Values

 -1 1 2 3 4 5 6 8 9 10 12 13 $$-1$$ $$1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{10}\right)$$
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)

$$\tau_{ a }( \chi_{ 77 }(2,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{77}(2,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(2,r) e\left(\frac{2r}{77}\right) = -8.7573733007+0.5553493243i$$

## Jacobi sum

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

$$J(\chi_{ 77 }(2,·),\chi_{ 77 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{77}(2,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(2,r) \chi_{77}(1,1-r) = 1$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 77 }(2,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{77}(2,·)) = \sum_{r \in \Z/77\Z} \chi_{77}(2,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right) = 6.5526872458+7.277496461i$$