# Properties

 Label 441.211 Modulus $441$ Conductor $441$ Order $21$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

H = DirichletGroup(441, base_ring=CyclotomicField(42))

M = H._module

chi = DirichletCharacter(H, M([14,30]))

pari: [g,chi] = znchar(Mod(211,441))

## Basic properties

 Modulus: $$441$$ Conductor: $$441$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$21$$ 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 441.ba

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: $$\Q(\zeta_{21})$$ Fixed field: Number field defined by a degree 21 polynomial

## Values on generators

$$(344,199)$$ → $$(e\left(\frac{1}{3}\right),e\left(\frac{5}{7}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$\chi_{ 441 }(211, a)$$ $$1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 441 }(211,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 441 }(211,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 441 }(211,·),\chi_{ 441 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 441 }(211,·)) \;$$ at $$\; a,b =$$ e.g. 1,2