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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(270, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([34,9]))
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pari:[g,chi] = znchar(Mod(257,270))
\(\chi_{270}(23,\cdot)\)
\(\chi_{270}(47,\cdot)\)
\(\chi_{270}(77,\cdot)\)
\(\chi_{270}(83,\cdot)\)
\(\chi_{270}(113,\cdot)\)
\(\chi_{270}(137,\cdot)\)
\(\chi_{270}(167,\cdot)\)
\(\chi_{270}(173,\cdot)\)
\(\chi_{270}(203,\cdot)\)
\(\chi_{270}(227,\cdot)\)
\(\chi_{270}(257,\cdot)\)
\(\chi_{270}(263,\cdot)\)
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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 ]
\((191,217)\) → \((e\left(\frac{17}{18}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 270 }(257, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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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)
