# Properties

 Modulus 87 Conductor 29 Order 28 Real no Primitive no Minimal yes Parity odd Orbit label 87.l

# Learn more about

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

sage: H = DirichletGroup(87)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,23]))

pari: [g,chi] = znchar(Mod(10,87))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 87 Conductor = 29 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 28 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 87.l Orbit index = 12

## Galois orbit

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Values on generators

$$(59,31)$$ → $$(1,e\left(\frac{23}{28}\right))$$

## Values

 -1 1 2 4 5 7 8 10 11 13 14 16 $$-1$$ $$1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{2}{7}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{28})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 87 }(10,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{87}(10,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(10,r) e\left(\frac{2r}{87}\right) = 4.594687749+-2.8087086872i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 87 }(10,·),\chi_{ 87 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{87}(10,\cdot),\chi_{87}(1,\cdot)) = \sum_{r\in \Z/87\Z} \chi_{87}(10,r) \chi_{87}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 87 }(10,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{87}(10,·)) = \sum_{r \in \Z/87\Z} \chi_{87}(10,r) e\left(\frac{1 r + 2 r^{-1}}{87}\right) = -7.9675885717+-12.6803445712i$$