Properties

 Label 170.69 Modulus $170$ Conductor $5$ Order $2$ Real yes Primitive no Minimal yes Parity even

Related objects

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

sage: H = DirichletGroup(170)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([1,0]))

pari: [g,chi] = znchar(Mod(69,170))

Basic properties

 Modulus: $$170$$ Conductor: $$5$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{5}(4,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 170.c

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

$$(137,71)$$ → $$(-1,1)$$

Values

 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$19$$ $$21$$ $$23$$ $$27$$ $$29$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$
 value at e.g. 2

Related number fields

 Field of values: $$\Q$$ Fixed field: $$\Q(\sqrt{5})$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 170 }(69,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{170}(69,\cdot)) = \sum_{r\in \Z/170\Z} \chi_{170}(69,r) e\left(\frac{r}{85}\right) = 2.2360679775$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 170 }(69,·),\chi_{ 170 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{170}(69,\cdot),\chi_{170}(1,\cdot)) = \sum_{r\in \Z/170\Z} \chi_{170}(69,r) \chi_{170}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 170 }(69,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{170}(69,·)) = \sum_{r \in \Z/170\Z} \chi_{170}(69,r) e\left(\frac{1 r + 2 r^{-1}}{170}\right) = 0.0$$