# Properties

 Label 145.6 Modulus $145$ Conductor $29$ Order $14$ Real no Primitive no Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(145, base_ring=CyclotomicField(14))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3]))

pari: [g,chi] = znchar(Mod(6,145))

## Basic properties

 Modulus: $$145$$ Conductor: $$29$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$14$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{29}(6,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 145.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: $$\Q(\zeta_{7})$$ Fixed field: $$\Q(\zeta_{29})^+$$

## Values on generators

$$(117,31)$$ → $$(1,e\left(\frac{3}{14}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$1$$ $$1$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$-1$$ $$e\left(\frac{6}{7}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 145 }(6,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{145}(6,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(6,r) e\left(\frac{2r}{145}\right) = 1.923871614+5.029783098i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 145 }(6,·),\chi_{ 145 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{145}(6,\cdot),\chi_{145}(1,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(6,r) \chi_{145}(1,1-r) = -3$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 145 }(6,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{145}(6,·)) = \sum_{r \in \Z/145\Z} \chi_{145}(6,r) e\left(\frac{1 r + 2 r^{-1}}{145}\right) = 5.5942277633+4.4612477726i$$