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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(248, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
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pari:[g,chi] = znchar(Mod(65,248))
\(\chi_{248}(17,\cdot)\)
\(\chi_{248}(65,\cdot)\)
\(\chi_{248}(73,\cdot)\)
\(\chi_{248}(105,\cdot)\)
\(\chi_{248}(137,\cdot)\)
\(\chi_{248}(145,\cdot)\)
\(\chi_{248}(177,\cdot)\)
\(\chi_{248}(241,\cdot)\)
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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 ]
\((63,125,65)\) → \((1,1,e\left(\frac{1}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 248 }(65, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) |
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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)
