# Properties

 Label 572.31 Modulus $572$ Conductor $572$ Order $20$ 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(572, base_ring=CyclotomicField(20))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([10,12,15]))

pari: [g,chi] = znchar(Mod(31,572))

## Basic properties

 Modulus: $$572$$ Conductor: $$572$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$20$$ 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 572.bk

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: $$\Q(\zeta_{20})$$ Fixed field: 20.20.2466227638053818291661582077909665841152.1

## Values on generators

$$(287,365,353)$$ → $$(-1,e\left(\frac{3}{5}\right),-i)$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$15$$ $$17$$ $$19$$ $$21$$ $$23$$ $$25$$ $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$i$$ $$1$$ $$e\left(\frac{3}{10}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 572 }(31,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{572}(31,\cdot)) = \sum_{r\in \Z/572\Z} \chi_{572}(31,r) e\left(\frac{r}{286}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 572 }(31,·),\chi_{ 572 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{572}(31,\cdot),\chi_{572}(1,\cdot)) = \sum_{r\in \Z/572\Z} \chi_{572}(31,r) \chi_{572}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 572 }(31,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{572}(31,·)) = \sum_{r \in \Z/572\Z} \chi_{572}(31,r) e\left(\frac{1 r + 2 r^{-1}}{572}\right) = -3.9084837648+-7.6708312974i$$