# Properties

 Label 403.196 Modulus $403$ Conductor $31$ Order $15$ Real no Primitive no Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(403, base_ring=CyclotomicField(30))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,14]))

pari: [g,chi] = znchar(Mod(196,403))

## Basic properties

 Modulus: $$403$$ Conductor: $$31$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$15$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{31}(10,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 403.bi

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: $$\Q(\zeta_{15})$$ Fixed field: $$\Q(\zeta_{31})^+$$

## Values on generators

$$(249,313)$$ → $$(1,e\left(\frac{7}{15}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 403 }(196,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{403}(196,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(196,r) e\left(\frac{2r}{403}\right) = 0.5780360736+-5.537677699i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 403 }(196,·),\chi_{ 403 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{403}(196,\cdot),\chi_{403}(1,\cdot)) = \sum_{r\in \Z/403\Z} \chi_{403}(196,r) \chi_{403}(1,1-r) = -11$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 403 }(196,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{403}(196,·)) = \sum_{r \in \Z/403\Z} \chi_{403}(196,r) e\left(\frac{1 r + 2 r^{-1}}{403}\right) = 34.2311116945+24.8703584269i$$