# Properties

 Modulus 103 Conductor 103 Order 51 Real no Primitive yes Minimal yes Parity even Orbit label 103.g

# Related objects

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

sage: H = DirichletGroup(103)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([49]))

pari: [g,chi] = znchar(Mod(59,103))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 103 Conductor = 103 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 51 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 = even Orbit label = 103.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

$$5$$ → $$e\left(\frac{49}{51}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{14}{51}\right)$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{28}{51}\right)$$ $$e\left(\frac{49}{51}\right)$$ $$e\left(\frac{38}{51}\right)$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{16}{17}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{31}{51}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{51})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 103 }(59,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{103}(59,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(59,r) e\left(\frac{2r}{103}\right) = 5.2698021331+-8.6734759744i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 103 }(59,·),\chi_{ 103 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{103}(59,\cdot),\chi_{103}(1,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(59,r) \chi_{103}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 103 }(59,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{103}(59,·)) = \sum_{r \in \Z/103\Z} \chi_{103}(59,r) e\left(\frac{1 r + 2 r^{-1}}{103}\right) = -0.1762684514+-0.2057414133i$$