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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(900, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,10,17]))
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pari:[g,chi] = znchar(Mod(197,900))
\(\chi_{900}(17,\cdot)\)
\(\chi_{900}(53,\cdot)\)
\(\chi_{900}(197,\cdot)\)
\(\chi_{900}(233,\cdot)\)
\(\chi_{900}(377,\cdot)\)
\(\chi_{900}(413,\cdot)\)
\(\chi_{900}(737,\cdot)\)
\(\chi_{900}(773,\cdot)\)
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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 ]
\((451,101,577)\) → \((1,-1,e\left(\frac{17}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 900 }(197, a) \) |
\(1\) | \(1\) | \(i\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
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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)
