# Properties

 Conductor 99 Order 30 Real No Primitive Yes Parity Odd Orbit Label 99.o

# Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(99)
sage: chi = H[40]
pari: [g,chi] = znchar(Mod(40,99))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 99 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 sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Odd Orbit label = 99.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

$$(56,46)$$ → $$(e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))$$

## Values

 -1 1 2 4 5 7 8 10 13 14 16 17 $$-1$$ $$1$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$-1$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{10}\right)$$
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)
$$\tau_{ a }( \chi_{ 99 }(40,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{99}(40,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(40,r) e\left(\frac{2r}{99}\right) = -9.9498739821+-0.0027822572i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 99 }(40,·),\chi_{ 99 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{99}(40,\cdot),\chi_{99}(1,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(40,r) \chi_{99}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 99 }(40,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{99}(40,·)) = \sum_{r \in \Z/99\Z} \chi_{99}(40,r) e\left(\frac{1 r + 2 r^{-1}}{99}\right) = 0.8932944415+-8.4991288813i$$