# Properties

 Label 103.13 Modulus $103$ Conductor $103$ Order $17$ 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(103, base_ring=CyclotomicField(34))

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(13,103))

## Basic properties

 Modulus: $$103$$ Conductor: $$103$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$17$$ 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 103.e

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: $$\Q(\zeta_{17})$$ Fixed field: 17.17.160470643909878751793805444097921.1

## Values on generators

$$5$$ → $$e\left(\frac{12}{17}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{2}{17}\right)$$ $$e\left(\frac{12}{17}\right)$$ $$e\left(\frac{10}{17}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{1}{17}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 103 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{103}(13,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(13,r) e\left(\frac{2r}{103}\right) = 8.6183233224+-5.3595245228i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 103 }(13,·),\chi_{ 103 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{103}(13,\cdot),\chi_{103}(1,\cdot)) = \sum_{r\in \Z/103\Z} \chi_{103}(13,r) \chi_{103}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 103 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{103}(13,·)) = \sum_{r \in \Z/103\Z} \chi_{103}(13,r) e\left(\frac{1 r + 2 r^{-1}}{103}\right) = -16.2545612475+-3.0385040979i$$