# Properties

 Label 712.145 Modulus $712$ Conductor $89$ Order $88$ Real no Primitive no Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(712, base_ring=CyclotomicField(88))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,0,41]))

pari: [g,chi] = znchar(Mod(145,712))

## Basic properties

 Modulus: $$712$$ Conductor: $$89$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$88$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{89}(56,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 712.be

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: $\Q(\zeta_{88})$ Fixed field: Number field defined by a degree 88 polynomial

## Values on generators

$$(535,357,537)$$ → $$(1,1,e\left(\frac{41}{88}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$\chi_{ 712 }(145, a)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 712 }(145,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 712 }(145,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 712 }(145,·),\chi_{ 712 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 712 }(145,·)) \;$$ at $$\; a,b =$$ e.g. 1,2