# Properties

 Label 264.197 Modulus $264$ Conductor $264$ Order $2$ Real yes Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(264, base_ring=CyclotomicField(2))

M = H._module

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

pari: [g,chi] = znchar(Mod(197,264))

## Kronecker symbol representation

sage: kronecker_character(264)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{264}{\bullet}\right)$$

## Basic properties

 Modulus: $$264$$ Conductor: $$264$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes 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 264.m

sage: chi.galois_orbit()

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{66})$$

## Values on generators

$$(199,133,89,145)$$ → $$(1,-1,-1,-1)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$7$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$ $$\chi_{ 264 }(197, a)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 264 }(197,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 264 }(197,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 264 }(197,·),\chi_{ 264 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 264 }(197,·)) \;$$ at $$\; a,b =$$ e.g. 1,2