# Properties

 Label 320.197 Modulus $320$ Conductor $320$ Order $16$ 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(320, base_ring=CyclotomicField(16))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,1,4]))

pari: [g,chi] = znchar(Mod(197,320))

## Basic properties

 Modulus: $$320$$ Conductor: $$320$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$16$$ 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 320.bi

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: $$\Q(\zeta_{16})$$ Fixed field: 16.0.147573952589676412928000000000000.2

## Values on generators

$$(191,261,257)$$ → $$(1,e\left(\frac{1}{16}\right),i)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$\chi_{ 320 }(197, a)$$ $$-1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 320 }(197,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 320 }(197,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 320 }(197,·),\chi_{ 320 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 320 }(197,·)) \;$$ at $$\; a,b =$$ e.g. 1,2