# Properties

 Label 145.27 Modulus $145$ Conductor $145$ Order $28$ 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(28))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([7,15]))

pari: [g,chi] = znchar(Mod(27,145))

## Basic properties

 Modulus: $$145$$ Conductor: $$145$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$28$$ 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.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: $$\Q(\zeta_{28})$$ Fixed field: 28.28.1455848000512373226044338588471370773272037506103515625.2

## Values on generators

$$(117,31)$$ → $$(i,e\left(\frac{15}{28}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$1$$ $$1$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$1$$ $$e\left(\frac{11}{28}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 145 }(27,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{145}(27,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(27,r) e\left(\frac{2r}{145}\right) = -11.5362039657+-3.4519556865i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 145 }(27,·),\chi_{ 145 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{145}(27,\cdot),\chi_{145}(1,\cdot)) = \sum_{r\in \Z/145\Z} \chi_{145}(27,r) \chi_{145}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 145 }(27,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{145}(27,·)) = \sum_{r \in \Z/145\Z} \chi_{145}(27,r) e\left(\frac{1 r + 2 r^{-1}}{145}\right) = 4.6319363443+-3.6938459736i$$