# Properties

 Conductor 181 Order 45 Real No Primitive Yes Parity Even Orbit Label 181.o

# 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(181)
sage: chi = H[16]
pari: [g,chi] = znchar(Mod(16,181))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 181 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 45 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 = Even Orbit label = 181.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

$$2$$ → $$e\left(\frac{1}{45}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{45})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 181 }(16,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{181}(16,\cdot)) = \sum_{r\in \Z/181\Z} \chi_{181}(16,r) e\left(\frac{2r}{181}\right) = -4.8210066303+12.5601709809i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 181 }(16,·),\chi_{ 181 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{181}(16,\cdot),\chi_{181}(1,\cdot)) = \sum_{r\in \Z/181\Z} \chi_{181}(16,r) \chi_{181}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 181 }(16,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{181}(16,·)) = \sum_{r \in \Z/181\Z} \chi_{181}(16,r) e\left(\frac{1 r + 2 r^{-1}}{181}\right) = -10.853848629+-0.7589750319i$$