# Properties

 Label 137.127 Modulus $137$ Conductor $137$ Order $8$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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([1]))

pari: [g,chi] = znchar(Mod(127,137))

## Basic properties

 Modulus: $$137$$ Conductor: $$137$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 137.d

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

$$3$$ → $$e\left(\frac{1}{8}\right)$$

## Values

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

## Related number fields

 Field of values: $$\Q(\zeta_{8})$$ Fixed field: 8.0.905824306333433.1

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 137 }(127,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{137}(127,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(127,r) e\left(\frac{2r}{137}\right) = -6.3024828215+-9.8629970235i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 137 }(127,·),\chi_{ 137 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{137}(127,\cdot),\chi_{137}(1,\cdot)) = \sum_{r\in \Z/137\Z} \chi_{137}(127,r) \chi_{137}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 137 }(127,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{137}(127,·)) = \sum_{r \in \Z/137\Z} \chi_{137}(127,r) e\left(\frac{1 r + 2 r^{-1}}{137}\right) = -4.4589429105+4.4589429105i$$