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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(385, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,40,54]))
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pari:[g,chi] = znchar(Mod(193,385))
| Modulus: | \(385\) | |
| Conductor: | \(385\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(60\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{385}(2,\cdot)\)
\(\chi_{385}(18,\cdot)\)
\(\chi_{385}(72,\cdot)\)
\(\chi_{385}(107,\cdot)\)
\(\chi_{385}(123,\cdot)\)
\(\chi_{385}(128,\cdot)\)
\(\chi_{385}(172,\cdot)\)
\(\chi_{385}(193,\cdot)\)
\(\chi_{385}(228,\cdot)\)
\(\chi_{385}(233,\cdot)\)
\(\chi_{385}(277,\cdot)\)
\(\chi_{385}(282,\cdot)\)
\(\chi_{385}(303,\cdot)\)
\(\chi_{385}(338,\cdot)\)
\(\chi_{385}(347,\cdot)\)
\(\chi_{385}(382,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((232,276,211)\) → \((-i,e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 385 }(193, a) \) |
\(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
