# Properties

 Label 273.136 Modulus $273$ Conductor $91$ Order $12$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(273, base_ring=CyclotomicField(12))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,2,5]))

pari: [g,chi] = znchar(Mod(136,273))

## Basic properties

 Modulus: $$273$$ Conductor: $$91$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{91}(45,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 273.bt

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: $$\Q(\zeta_{12})$$ Fixed field: 12.12.506240953553539690213.2

## Values on generators

$$(92,157,106)$$ → $$(1,e\left(\frac{1}{6}\right),e\left(\frac{5}{12}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$16$$ $$17$$ $$19$$ $$20$$ $$1$$ $$1$$ $$-i$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 273 }(136,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{273}(136,\cdot)) = \sum_{r\in \Z/273\Z} \chi_{273}(136,r) e\left(\frac{2r}{273}\right) = 7.8094482181+5.478368236i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 273 }(136,·),\chi_{ 273 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{273}(136,\cdot),\chi_{273}(1,\cdot)) = \sum_{r\in \Z/273\Z} \chi_{273}(136,r) \chi_{273}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 273 }(136,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{273}(136,·)) = \sum_{r \in \Z/273\Z} \chi_{273}(136,r) e\left(\frac{1 r + 2 r^{-1}}{273}\right) = 18.6399132898+-18.6399132898i$$