# Properties

 Label 143.40 Modulus $143$ Conductor $11$ Order $10$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(143, base_ring=CyclotomicField(10))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([7,0]))

pari: [g,chi] = znchar(Mod(40,143))

## Basic properties

 Modulus: $$143$$ Conductor: $$11$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$10$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{11}(7,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 143.m

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\zeta_{5})$$ Fixed field: $$\Q(\zeta_{11})$$

## Values on generators

$$(79,67)$$ → $$(e\left(\frac{7}{10}\right),1)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$ $$-1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$-1$$ $$1$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 143 }(40,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{143}(40,\cdot)) = \sum_{r\in \Z/143\Z} \chi_{143}(40,r) e\left(\frac{2r}{143}\right) = -2.5412780247+2.1311747937i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 143 }(40,·),\chi_{ 143 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{143}(40,\cdot),\chi_{143}(1,\cdot)) = \sum_{r\in \Z/143\Z} \chi_{143}(40,r) \chi_{143}(1,1-r) = -11$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 143 }(40,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{143}(40,·)) = \sum_{r \in \Z/143\Z} \chi_{143}(40,r) e\left(\frac{1 r + 2 r^{-1}}{143}\right) = 2.006477495+1.4577912316i$$