# Properties

 Conductor 53 Order 52 Real no Primitive yes Minimal yes Parity odd Orbit label 53.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(53)

sage: chi = H[48]

pari: [g,chi] = znchar(Mod(48,53))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 53 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 52 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 53.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

$$2$$ → $$e\left(\frac{21}{52}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$-1$$ $$1$$ $$e\left(\frac{21}{52}\right)$$ $$e\left(\frac{45}{52}\right)$$ $$e\left(\frac{21}{26}\right)$$ $$e\left(\frac{51}{52}\right)$$ $$e\left(\frac{7}{26}\right)$$ $$e\left(\frac{17}{26}\right)$$ $$e\left(\frac{11}{52}\right)$$ $$e\left(\frac{19}{26}\right)$$ $$e\left(\frac{5}{13}\right)$$ $$e\left(\frac{11}{26}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{52})$$

## Gauss sum

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

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 53 }(48,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{53}(48,\cdot)) = \sum_{r\in \Z/53\Z} \chi_{53}(48,r) e\left(\frac{2r}{53}\right) = -4.0789300043+-6.0301185743i$$

## Jacobi sum

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

$$J(\chi_{ 53 }(48,·),\chi_{ 53 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{53}(48,\cdot),\chi_{53}(1,\cdot)) = \sum_{r\in \Z/53\Z} \chi_{53}(48,r) \chi_{53}(1,1-r) = -1$$

## Kloosterman sum

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

$$K(a,b,\chi_{ 53 }(48,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{53}(48,·)) = \sum_{r \in \Z/53\Z} \chi_{53}(48,r) e\left(\frac{1 r + 2 r^{-1}}{53}\right) = -4.5670977727+1.4231652227i$$