# Properties

 Conductor 37 Order 9 Real No Primitive No Parity Even Orbit Label 74.f

# 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(74)
sage: chi = H[49]
pari: [g,chi] = znchar(Mod(49,74))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 37 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 sage: chi.is_odd() pari: zncharisodd(g,chi) Parity = Even Orbit label = 74.f Orbit index = 6

## 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

$$39$$ → $$e\left(\frac{7}{9}\right)$$

## Values

 -1 1 3 5 7 9 11 13 15 17 19 21 $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$
value at  e.g. 2

## Related number fields

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

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 74 }(49,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{74}(49,\cdot)) = \sum_{r\in \Z/74\Z} \chi_{74}(49,r) e\left(\frac{r}{37}\right) = 4.391540734+-4.2088442573i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 74 }(49,·),\chi_{ 74 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{74}(49,\cdot),\chi_{74}(1,\cdot)) = \sum_{r\in \Z/74\Z} \chi_{74}(49,r) \chi_{74}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 74 }(49,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{74}(49,·)) = \sum_{r \in \Z/74\Z} \chi_{74}(49,r) e\left(\frac{1 r + 2 r^{-1}}{74}\right) = -6.6896798048+5.6133078569i$$