# Properties

 Label 143.107 Modulus $143$ Conductor $143$ Order $30$ Real no Primitive yes Minimal yes Parity odd

# Learn more about

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

sage: H = DirichletGroup(143, base_ring=CyclotomicField(30))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([9,10]))

pari: [g,chi] = znchar(Mod(107,143))

## Basic properties

 Modulus: $$143$$ Conductor: $$143$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$30$$ 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 143.t

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: $$\Q(\zeta_{15})$$ Fixed field: 30.0.249154964698353870876083085574129912384252301255171.1

## Values on generators

$$(79,67)$$ → $$(e\left(\frac{3}{10}\right),e\left(\frac{1}{3}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$ $$-1$$ $$1$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 143 }(107,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{143}(107,\cdot)) = \sum_{r\in \Z/143\Z} \chi_{143}(107,r) e\left(\frac{2r}{143}\right) = 5.1201440001+-10.8066704131i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 143 }(107,·),\chi_{ 143 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{143}(107,\cdot),\chi_{143}(1,\cdot)) = \sum_{r\in \Z/143\Z} \chi_{143}(107,r) \chi_{143}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 143 }(107,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{143}(107,·)) = \sum_{r \in \Z/143\Z} \chi_{143}(107,r) e\left(\frac{1 r + 2 r^{-1}}{143}\right) = -14.9585644508+-6.6599819845i$$