# Properties

 Conductor 235 Order 92 Real No Primitive Yes Parity Even Orbit Label 235.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(235)
sage: chi = H[38]
pari: [g,chi] = znchar(Mod(38,235))

## Basic properties

 sage: chi.conductor() pari: znconreyconductor(g,chi) Conductor = 235 sage: chi.multiplicative_order() pari: charorder(g,chi) Order = 92 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 = 235.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

$$(142,146)$$ → $$(-i,e\left(\frac{17}{46}\right))$$

## Values

 -1 1 2 3 4 6 7 8 9 11 12 13 $$1$$ $$1$$ $$e\left(\frac{37}{92}\right)$$ $$e\left(\frac{59}{92}\right)$$ $$e\left(\frac{37}{46}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{53}{92}\right)$$ $$e\left(\frac{19}{92}\right)$$ $$e\left(\frac{13}{46}\right)$$ $$e\left(\frac{27}{46}\right)$$ $$e\left(\frac{41}{92}\right)$$ $$e\left(\frac{29}{92}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{92})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
$$\tau_{ a }( \chi_{ 235 }(38,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{235}(38,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(38,r) e\left(\frac{2r}{235}\right) = 10.7796533312+-10.8994987985i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
$$J(\chi_{ 235 }(38,·),\chi_{ 235 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{235}(38,\cdot),\chi_{235}(1,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(38,r) \chi_{235}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
$$K(a,b,\chi_{ 235 }(38,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{235}(38,·)) = \sum_{r \in \Z/235\Z} \chi_{235}(38,r) e\left(\frac{1 r + 2 r^{-1}}{235}\right) = 4.9232920047+15.5120142025i$$