# Properties

 Label 114.37 Modulus $114$ Conductor $19$ Order $2$ Real yes Primitive no Minimal yes Parity odd

# Learn more

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

sage: H = DirichletGroup(114, base_ring=CyclotomicField(2))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(37,114))

## Basic properties

 Modulus: $$114$$ Conductor: $$19$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{19}(18,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 114.d

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{-19})$$

## Values on generators

$$(77,97)$$ → $$(1,-1)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$ $$\chi_{ 114 }(37, a)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 114 }(37,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 114 }(37,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 114 }(37,·),\chi_{ 114 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 114 }(37,·)) \;$$ at $$\; a,b =$$ e.g. 1,2