# Properties

 Conductor 73 Order 72 Real No Primitive Yes Parity Odd Orbit Label 73.l

# 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(73)
sage: chi = H[40]
pari: [g,chi] = znchar(Mod(40,73))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 73 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 72 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 = 73.l Orbit index = 12

## 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

$$5$$ → $$e\left(\frac{25}{72}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{25}{72}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{72}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{72})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 73 }(40,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{73}(40,\cdot)) = \sum_{r\in \Z/73\Z} \chi_{73}(40,r) e\left(\frac{2r}{73}\right) = -3.9936425916+-7.5531992461i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 73 }(40,·),\chi_{ 73 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{73}(40,\cdot),\chi_{73}(1,\cdot)) = \sum_{r\in \Z/73\Z} \chi_{73}(40,r) \chi_{73}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 73 }(40,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{73}(40,·)) = \sum_{r \in \Z/73\Z} \chi_{73}(40,r) e\left(\frac{1 r + 2 r^{-1}}{73}\right) = -3.7606671186+-4.4817885491i$$