Properties

 Label 709.25 Modulus $709$ Conductor $709$ Order $177$ 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(709)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([161]))

pari: [g,chi] = znchar(Mod(25,709))

Basic properties

 Modulus: $$709$$ Conductor: $$709$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$177$$ 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 709.i

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

$$2$$ → $$e\left(\frac{161}{177}\right)$$

Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{161}{177}\right)$$ $$e\left(\frac{80}{177}\right)$$ $$e\left(\frac{145}{177}\right)$$ $$e\left(\frac{158}{177}\right)$$ $$e\left(\frac{64}{177}\right)$$ $$e\left(\frac{53}{177}\right)$$ $$e\left(\frac{43}{59}\right)$$ $$e\left(\frac{160}{177}\right)$$ $$e\left(\frac{142}{177}\right)$$ $$e\left(\frac{77}{177}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $\Q(\zeta_{177})$ Fixed field: Number field defined by a degree 177 polynomial

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 709 }(25,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{709}(25,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(25,r) e\left(\frac{2r}{709}\right) = 25.7777211779+-6.671513387i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 709 }(25,·),\chi_{ 709 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{709}(25,\cdot),\chi_{709}(1,\cdot)) = \sum_{r\in \Z/709\Z} \chi_{709}(25,r) \chi_{709}(1,1-r) = -1$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 709 }(25,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{709}(25,·)) = \sum_{r \in \Z/709\Z} \chi_{709}(25,r) e\left(\frac{1 r + 2 r^{-1}}{709}\right) = -5.7063322988+1.6655337466i$$