# Properties

 Label 503.2 Modulus $503$ Conductor $503$ Order $251$ Real no Primitive yes Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(503, base_ring=CyclotomicField(502))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([202]))

pari: [g,chi] = znchar(Mod(2,503))

## Basic properties

 Modulus: $$503$$ Conductor: $$503$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$251$$ 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 503.c

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

$$5$$ → $$e\left(\frac{101}{251}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{71}{251}\right)$$ $$e\left(\frac{194}{251}\right)$$ $$e\left(\frac{142}{251}\right)$$ $$e\left(\frac{101}{251}\right)$$ $$e\left(\frac{14}{251}\right)$$ $$e\left(\frac{152}{251}\right)$$ $$e\left(\frac{213}{251}\right)$$ $$e\left(\frac{137}{251}\right)$$ $$e\left(\frac{172}{251}\right)$$ $$e\left(\frac{226}{251}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{251})$ Fixed field: Number field defined by a degree 251 polynomial (not computed)

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 503 }(2,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{503}(2,\cdot)) = \sum_{r\in \Z/503\Z} \chi_{503}(2,r) e\left(\frac{2r}{503}\right) = -2.8483021926+22.2460597549i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 503 }(2,·),\chi_{ 503 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{503}(2,\cdot),\chi_{503}(1,\cdot)) = \sum_{r\in \Z/503\Z} \chi_{503}(2,r) \chi_{503}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 503 }(2,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{503}(2,·)) = \sum_{r \in \Z/503\Z} \chi_{503}(2,r) e\left(\frac{1 r + 2 r^{-1}}{503}\right) = 4.1103917765+5.060783155i$$