# Properties

 Label 147.64 Modulus $147$ Conductor $49$ Order $7$ 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(147, base_ring=CyclotomicField(14))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,10]))

pari: [g,chi] = znchar(Mod(64,147))

## Basic properties

 Modulus: $$147$$ Conductor: $$49$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$7$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{49}(15,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 147.i

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: $$\Q(\zeta_{7})$$ Fixed field: 7.7.13841287201.1

## Values on generators

$$(50,52)$$ → $$(1,e\left(\frac{5}{7}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$1$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 147 }(64,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{147}(64,\cdot)) = \sum_{r\in \Z/147\Z} \chi_{147}(64,r) e\left(\frac{2r}{147}\right) = -0.224361043+6.9964035134i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 147 }(64,·),\chi_{ 147 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{147}(64,\cdot),\chi_{147}(1,\cdot)) = \sum_{r\in \Z/147\Z} \chi_{147}(64,r) \chi_{147}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 147 }(64,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{147}(64,·)) = \sum_{r \in \Z/147\Z} \chi_{147}(64,r) e\left(\frac{1 r + 2 r^{-1}}{147}\right) = 0.5982797354+-2.6212347887i$$