Properties

 Label 199.140 Modulus $199$ Conductor $199$ Order $33$ 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(199, base_ring=CyclotomicField(66))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([32]))

pari: [g,chi] = znchar(Mod(140,199))

Basic properties

 Modulus: $$199$$ Conductor: $$199$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$33$$ 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 199.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

$$3$$ → $$e\left(\frac{16}{33}\right)$$

Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$1$$ $$1$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{7}{11}\right)$$
 value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 199 }(140,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{199}(140,\cdot)) = \sum_{r\in \Z/199\Z} \chi_{199}(140,r) e\left(\frac{2r}{199}\right) = -11.9657154835+7.4713889585i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 199 }(140,·),\chi_{ 199 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{199}(140,\cdot),\chi_{199}(1,\cdot)) = \sum_{r\in \Z/199\Z} \chi_{199}(140,r) \chi_{199}(1,1-r) = -1$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 199 }(140,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{199}(140,·)) = \sum_{r \in \Z/199\Z} \chi_{199}(140,r) e\left(\frac{1 r + 2 r^{-1}}{199}\right) = 1.3990689712+4.0423443184i$$