# Properties

 Label 775.133 Modulus $775$ Conductor $775$ Order $60$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(775, base_ring=CyclotomicField(60))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([9,4]))

pari: [g,chi] = znchar(Mod(133,775))

## Basic properties

 Modulus: $$775$$ Conductor: $$775$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$60$$ 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 775.cv

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: $$\Q(\zeta_{60})$$ Fixed field: Number field defined by a degree 60 polynomial

## Values on generators

$$(652,251)$$ → $$(e\left(\frac{3}{20}\right),e\left(\frac{1}{15}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$\chi_{ 775 }(133, a)$$ $$-1$$ $$1$$ $$-i$$ $$e\left(\frac{7}{60}\right)$$ $$-1$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$i$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{7}{12}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 775 }(133,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 775 }(133,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 775 }(133,·),\chi_{ 775 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 775 }(133,·)) \;$$ at $$\; a,b =$$ e.g. 1,2