# Properties

 Label 916.501 Modulus $916$ Conductor $229$ Order $19$ 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(916, base_ring=CyclotomicField(38))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,24]))

pari: [g,chi] = znchar(Mod(501,916))

## Basic properties

 Modulus: $$916$$ Conductor: $$229$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$19$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{229}(43,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 916.m

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: $$\Q(\zeta_{19})$$ Fixed field: 19.19.2999429662895796650415561622892044448017561.1

## Values on generators

$$(459,693)$$ → $$(1,e\left(\frac{12}{19}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$1$$ $$1$$ $$e\left(\frac{7}{19}\right)$$ $$e\left(\frac{17}{19}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{6}{19}\right)$$ $$e\left(\frac{17}{19}\right)$$ $$e\left(\frac{5}{19}\right)$$ $$e\left(\frac{17}{19}\right)$$ $$e\left(\frac{2}{19}\right)$$ $$e\left(\frac{18}{19}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 916 }(501,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{916}(501,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(501,r) e\left(\frac{r}{458}\right) = 24.1937543808+-18.1841207915i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 916 }(501,·),\chi_{ 916 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{916}(501,\cdot),\chi_{916}(1,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(501,r) \chi_{916}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 916 }(501,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{916}(501,·)) = \sum_{r \in \Z/916\Z} \chi_{916}(501,r) e\left(\frac{1 r + 2 r^{-1}}{916}\right) = -0.0$$