# Properties

 Label 153.125 Modulus $153$ Conductor $51$ Order $16$ Real no Primitive no Minimal no Parity even

# Related objects

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

H = DirichletGroup(153, base_ring=CyclotomicField(16))

M = H._module

chi = DirichletCharacter(H, M([8,15]))

pari: [g,chi] = znchar(Mod(125,153))

## Basic properties

 Modulus: $$153$$ Conductor: $$51$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$16$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{51}(23,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 153.o

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: $$\Q(\zeta_{16})$$ Fixed field: $$\Q(\zeta_{51})^+$$

## Values on generators

$$(137,37)$$ → $$(-1,e\left(\frac{15}{16}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$\chi_{ 153 }(125, a)$$ $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 153 }(125,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 153 }(125,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 153 }(125,·),\chi_{ 153 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 153 }(125,·)) \;$$ at $$\; a,b =$$ e.g. 1,2