# Properties

 Modulus 137 Conductor 137 Order 34 Real no Primitive yes Minimal yes Parity even Orbit label 137.f

# Learn more about

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

sage: H = DirichletGroup(137)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([5]))

pari: [g,chi] = znchar(Mod(4,137))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 137 Conductor = 137 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 34 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 = 137.f Orbit index = 6

## 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

$$3$$ → $$e\left(\frac{5}{34}\right)$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$e\left(\frac{8}{17}\right)$$ $$e\left(\frac{5}{34}\right)$$ $$e\left(\frac{16}{17}\right)$$ $$e\left(\frac{1}{34}\right)$$ $$e\left(\frac{21}{34}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{5}{17}\right)$$ $$-1$$ $$e\left(\frac{16}{17}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{17})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 137 }(4,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{137}(4,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(4,r) e\left(\frac{2r}{137}\right) = -2.1772370071+-11.50041908i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 137 }(4,·),\chi_{ 137 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{137}(4,\cdot),\chi_{137}(1,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(4,r) \chi_{137}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 137 }(4,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{137}(4,·)) = \sum_{r \in \Z/137\Z} \chi_{137}(4,r) e\left(\frac{1 r + 2 r^{-1}}{137}\right) = 1.1134561996+12.0161060436i$$