Properties

 Label 344.27 Modulus $344$ Conductor $344$ Order $14$ Real no Primitive yes Minimal yes Parity even

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

sage: H = DirichletGroup(344)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([7,7,1]))

pari: [g,chi] = znchar(Mod(27,344))

Basic properties

 Modulus: $$344$$ Conductor: $$344$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$14$$ 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 344.v

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

$$(87,173,89)$$ → $$(-1,-1,e\left(\frac{1}{14}\right))$$

Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$1$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{14}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $$\Q(\zeta_{7})$$ Fixed field: 14.14.3603461044766854684853927936.1

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 344 }(27,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{344}(27,\cdot)) = \sum_{r\in \Z/344\Z} \chi_{344}(27,r) e\left(\frac{r}{172}\right) = -0.0$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 344 }(27,·),\chi_{ 344 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{344}(27,\cdot),\chi_{344}(1,\cdot)) = \sum_{r\in \Z/344\Z} \chi_{344}(27,r) \chi_{344}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 344 }(27,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{344}(27,·)) = \sum_{r \in \Z/344\Z} \chi_{344}(27,r) e\left(\frac{1 r + 2 r^{-1}}{344}\right) = -19.6078836369+4.4753714867i$$