# Properties

 Conductor 97 Order 96 Real No Primitive Yes Parity Odd Orbit Label 97.l

# 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(97)
sage: chi = H[21]
pari: [g,chi] = znchar(Mod(21,97))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 97 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 96 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 = 97.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

$$5$$ → $$e\left(\frac{5}{96}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{5}{96}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{59}{96}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{79}{96}\right)$$ $$e\left(\frac{23}{48}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{96})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 97 }(21,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{97}(21,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(21,r) e\left(\frac{2r}{97}\right) = -6.2409317734+-7.6191056299i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 97 }(21,·),\chi_{ 97 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{97}(21,\cdot),\chi_{97}(1,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(21,r) \chi_{97}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 97 }(21,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{97}(21,·)) = \sum_{r \in \Z/97\Z} \chi_{97}(21,r) e\left(\frac{1 r + 2 r^{-1}}{97}\right) = 3.1781728904+3.6240116178i$$