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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,29,0]))
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pari:[g,chi] = znchar(Mod(85,384))
\(\chi_{384}(13,\cdot)\)
\(\chi_{384}(37,\cdot)\)
\(\chi_{384}(61,\cdot)\)
\(\chi_{384}(85,\cdot)\)
\(\chi_{384}(109,\cdot)\)
\(\chi_{384}(133,\cdot)\)
\(\chi_{384}(157,\cdot)\)
\(\chi_{384}(181,\cdot)\)
\(\chi_{384}(205,\cdot)\)
\(\chi_{384}(229,\cdot)\)
\(\chi_{384}(253,\cdot)\)
\(\chi_{384}(277,\cdot)\)
\(\chi_{384}(301,\cdot)\)
\(\chi_{384}(325,\cdot)\)
\(\chi_{384}(349,\cdot)\)
\(\chi_{384}(373,\cdot)\)
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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 ]
\((127,133,257)\) → \((1,e\left(\frac{29}{32}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 384 }(85, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(i\) |
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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)
