# Properties

 Conductor 95 Order 12 Real No Primitive Yes Parity Odd Orbit Label 95.m

# 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(95)
sage: chi = H[87]
pari: [g,chi] = znchar(Mod(87,95))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 95 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 12 Real = No sage: chi.is_primitive() pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Orbit label = 95.m Orbit index = 13

## 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

$$(77,21)$$ → $$(i,e\left(\frac{2}{3}\right))$$

## Values

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

## Related number fields

 Field of values $$\Q(\zeta_{12})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 95 }(87,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{95}(87,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(87,r) e\left(\frac{2r}{95}\right) = -2.3436109247+9.4608397002i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 95 }(87,·),\chi_{ 95 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{95}(87,\cdot),\chi_{95}(1,\cdot)) = \sum_{r\in \Z/95\Z} \chi_{95}(87,r) \chi_{95}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 95 }(87,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{95}(87,·)) = \sum_{r \in \Z/95\Z} \chi_{95}(87,r) e\left(\frac{1 r + 2 r^{-1}}{95}\right) = -2.4632777504+-9.1930777177i$$