# Properties

 Label 77.53 Modulus $77$ Conductor $77$ Order $15$ 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(77, base_ring=CyclotomicField(30))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([20,18]))

pari: [g,chi] = znchar(Mod(53,77))

## Basic properties

 Modulus: $$77$$ Conductor: $$77$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$15$$ 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 77.m

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: 15.15.886528337182930278529.1

## Values on generators

$$(45,57)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{3}{5}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$12$$ $$13$$ $$1$$ $$1$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{3}{5}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 77 }(53,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{77}(53,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(53,r) e\left(\frac{2r}{77}\right) = 5.7558197842+-6.6234838727i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 77 }(53,·),\chi_{ 77 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{77}(53,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(53,r) \chi_{77}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 77 }(53,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{77}(53,·)) = \sum_{r \in \Z/77\Z} \chi_{77}(53,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right) = 4.9221449043+-1.0462341969i$$