# Properties

 Label 400.307 Modulus $400$ Conductor $80$ Order $4$ Real no Primitive no Minimal yes Parity even

# Related objects

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

H = DirichletGroup(400, base_ring=CyclotomicField(4))

M = H._module

chi = DirichletCharacter(H, M([2,3,1]))

pari: [g,chi] = znchar(Mod(307,400))

## Basic properties

 Modulus: $$400$$ Conductor: $$80$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$4$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{80}(67,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 400.j

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: $$\mathbb{Q}(i)$$ Fixed field: 4.4.256000.2

## Values on generators

$$(351,101,177)$$ → $$(-1,-i,i)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$\chi_{ 400 }(307, a)$$ $$1$$ $$1$$ $$-1$$ $$i$$ $$1$$ $$i$$ $$1$$ $$i$$ $$i$$ $$-i$$ $$-i$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 400 }(307,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 400 }(307,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 400 }(307,·),\chi_{ 400 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 400 }(307,·)) \;$$ at $$\; a,b =$$ e.g. 1,2