Properties

 Modulus 175 Conductor 175 Order 60 Real no Primitive yes Minimal yes Parity even Orbit label 175.x

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

sage: H = DirichletGroup(175)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([3,50]))

pari: [g,chi] = znchar(Mod(152,175))

Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 175 Conductor = 175 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 60 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 175.x Orbit index = 24

Galois orbit

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

$$(127,101)$$ → $$(e\left(\frac{1}{20}\right),e\left(\frac{5}{6}\right))$$

Values

 -1 1 2 3 4 6 8 9 11 12 13 16 $$1$$ $$1$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{13}{15}\right)$$
value at  e.g. 2

Related number fields

 Field of values $$\Q(\zeta_{60})$$

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 175 }(152,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{175}(152,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(152,r) e\left(\frac{2r}{175}\right) = -10.6938327622+7.7872935514i$$

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 175 }(152,·),\chi_{ 175 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{175}(152,\cdot),\chi_{175}(1,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(152,r) \chi_{175}(1,1-r) = 0$$

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 175 }(152,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{175}(152,·)) = \sum_{r \in \Z/175\Z} \chi_{175}(152,r) e\left(\frac{1 r + 2 r^{-1}}{175}\right) = 20.2927388466+-25.0594455i$$