# Properties

 Label 145.17 Modulus $145$ Conductor $145$ Order $4$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(145, base_ring=CyclotomicField(4))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,3]))

pari: [g,chi] = znchar(Mod(17,145))

## Basic properties

 Modulus: $$145$$ Conductor: $$145$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 145.j

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: $$\Q(\sqrt{-1})$$ Fixed field: 4.4.3048625.1

## Values on generators

$$(117,31)$$ → $$(i,-i)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$-i$$ $$-1$$ $$i$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 145 }(17,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{145}(17,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(17,r) e\left(\frac{2r}{145}\right) = 11.255539774+4.2793485948i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 145 }(17,·),\chi_{ 145 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{145}(17,\cdot),\chi_{145}(1,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(17,r) \chi_{145}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 145 }(17,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{145}(17,·)) = \sum_{r \in \Z/145\Z} \chi_{145}(17,r) e\left(\frac{1 r + 2 r^{-1}}{145}\right) = 16.7697733886$$