# Properties

 Conductor 373 Order 186 Real No Primitive Yes Parity Even Orbit Label 373.k

# 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(373)

sage: chi = H[59]

pari: [g,chi] = znchar(Mod(59,373))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 373 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 186 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 = 373.k Orbit index = 11

## 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

$$2$$ → $$e\left(\frac{173}{186}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{173}{186}\right)$$ $$e\left(\frac{34}{93}\right)$$ $$e\left(\frac{80}{93}\right)$$ $$e\left(\frac{35}{186}\right)$$ $$e\left(\frac{55}{186}\right)$$ $$e\left(\frac{5}{31}\right)$$ $$e\left(\frac{49}{62}\right)$$ $$e\left(\frac{68}{93}\right)$$ $$e\left(\frac{11}{93}\right)$$ $$e\left(\frac{163}{186}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{93})$$

## Gauss sum

sage: chi.sage_character().gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 373 }(59,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{373}(59,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(59,r) e\left(\frac{2r}{373}\right) = 12.2676989577+14.9165532977i$$

## Jacobi sum

sage: chi.sage_character().jacobi_sum(n)

$$J(\chi_{ 373 }(59,·),\chi_{ 373 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{373}(59,\cdot),\chi_{373}(1,\cdot)) = \sum_{r\in \Z/373\Z} \chi_{373}(59,r) \chi_{373}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)

$$K(a,b,\chi_{ 373 }(59,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{373}(59,·)) = \sum_{r \in \Z/373\Z} \chi_{373}(59,r) e\left(\frac{1 r + 2 r^{-1}}{373}\right) = 15.8709488133+-3.5419486246i$$