# Properties

 Modulus 55 Conductor 11 Order 5 Real no Primitive no Minimal yes Parity even Orbit label 55.g

# Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(55)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,1]))

pari: [g,chi] = znchar(Mod(26,55))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 55 Conductor = 11 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 5 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 55.g Orbit index = 7

## Galois orbit

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(12,46)$$ → $$(1,e\left(\frac{1}{5}\right))$$

## Values

 -1 1 2 3 4 6 7 8 9 12 13 14 $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{5})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 55 }(26,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{55}(26,\cdot)) = \sum_{r\in \Z/55\Z} \chi_{55}(26,r) e\left(\frac{2r}{55}\right) = 0.9496208436+3.177769698i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 55 }(26,·),\chi_{ 55 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{55}(26,\cdot),\chi_{55}(1,\cdot)) = \sum_{r\in \Z/55\Z} \chi_{55}(26,r) \chi_{55}(1,1-r) = -3$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 55 }(26,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{55}(26,·)) = \sum_{r \in \Z/55\Z} \chi_{55}(26,r) e\left(\frac{1 r + 2 r^{-1}}{55}\right) = 14.7093341188+10.6869567959i$$