# Properties

 Modulus 95 Conductor 19 Order 9 Real no Primitive no Minimal yes Parity even Orbit label 95.k

# Related objects

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

sage: H = DirichletGroup(95)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,1]))

pari: [g,chi] = znchar(Mod(61,95))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 95 Conductor = 19 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 9 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 95.k Orbit index = 11

## 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

$$(77,21)$$ → $$(1,e\left(\frac{1}{9}\right))$$

## Values

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

## Related number fields

 Field of values $$\Q(\zeta_{9})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 95 }(61,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{95}(61,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(61,r) e\left(\frac{2r}{95}\right) = -1.3301051539+4.1510023223i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 95 }(61,·),\chi_{ 95 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{95}(61,\cdot),\chi_{95}(1,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(61,r) \chi_{95}(1,1-r) = -3$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 95 }(61,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{95}(61,·)) = \sum_{r \in \Z/95\Z} \chi_{95}(61,r) e\left(\frac{1 r + 2 r^{-1}}{95}\right) = 15.2496101204+5.550404168i$$