# Properties

 Label 400.209 Modulus $400$ Conductor $25$ Order $10$ Real no Primitive no Minimal no Parity even

# Learn more

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

H = DirichletGroup(400, base_ring=CyclotomicField(10))

M = H._module

chi = DirichletCharacter(H, M([0,0,7]))

pari: [g,chi] = znchar(Mod(209,400))

## Basic properties

 Modulus: $$400$$ Conductor: $$25$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$10$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{25}(9,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 400.y

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: $$\Q(\zeta_{5})$$ Fixed field: $$\Q(\zeta_{25})^+$$

## Values on generators

$$(351,101,177)$$ → $$(1,1,e\left(\frac{7}{10}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$\chi_{ 400 }(209, a)$$ $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$-1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 400 }(209,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 400 }(209,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 400 }(209,·),\chi_{ 400 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 400 }(209,·)) \;$$ at $$\; a,b =$$ e.g. 1,2