# Properties

 Label 91.19 Modulus $91$ Conductor $91$ Order $12$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(91, base_ring=CyclotomicField(12))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([10,5]))

pari: [g,chi] = znchar(Mod(19,91))

## Basic properties

 Modulus: $$91$$ Conductor: $$91$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 91.w

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: 12.12.506240953553539690213.1

## Values on generators

$$(66,15)$$ → $$(e\left(\frac{5}{6}\right),e\left(\frac{5}{12}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$1$$ $$1$$ $$i$$ $$e\left(\frac{2}{3}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 91 }(19,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{91}(19,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(19,r) e\left(\frac{2r}{91}\right) = 8.9338052934+3.34471568i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 91 }(19,·),\chi_{ 91 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{91}(19,\cdot),\chi_{91}(1,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(19,r) \chi_{91}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 91 }(19,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{91}(19,·)) = \sum_{r \in \Z/91\Z} \chi_{91}(19,r) e\left(\frac{1 r + 2 r^{-1}}{91}\right) = 0.6339694245+0.1698715953i$$