# Properties

 Label 930.13 Modulus $930$ Conductor $155$ Order $60$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(930, base_ring=CyclotomicField(60))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,45,22]))

pari: [g,chi] = znchar(Mod(13,930))

## Basic properties

 Modulus: $$930$$ Conductor: $$155$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$60$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{155}(13,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 930.bt

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

$$(311,187,871)$$ → $$(1,-i,e\left(\frac{11}{30}\right))$$

## Values

 $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$37$$ $$41$$ $$43$$ $$1$$ $$1$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{13}{60}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{60})$$ Fixed field: Number field defined by a degree 60 polynomial

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 930 }(13,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{930}(13,\cdot)) = \sum_{r\in \Z/930\Z} \chi_{930}(13,r) e\left(\frac{r}{465}\right) = 8.3065483571+9.2736861275i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 930 }(13,·),\chi_{ 930 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{930}(13,\cdot),\chi_{930}(1,\cdot)) = \sum_{r\in \Z/930\Z} \chi_{930}(13,r) \chi_{930}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 930 }(13,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{930}(13,·)) = \sum_{r \in \Z/930\Z} \chi_{930}(13,r) e\left(\frac{1 r + 2 r^{-1}}{930}\right) = 3.2689873334+-20.6395737278i$$