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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(676, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([0,59]))
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pari:[g,chi] = znchar(Mod(309,676))
\(\chi_{676}(17,\cdot)\)
\(\chi_{676}(49,\cdot)\)
\(\chi_{676}(69,\cdot)\)
\(\chi_{676}(101,\cdot)\)
\(\chi_{676}(121,\cdot)\)
\(\chi_{676}(153,\cdot)\)
\(\chi_{676}(173,\cdot)\)
\(\chi_{676}(205,\cdot)\)
\(\chi_{676}(225,\cdot)\)
\(\chi_{676}(257,\cdot)\)
\(\chi_{676}(277,\cdot)\)
\(\chi_{676}(309,\cdot)\)
\(\chi_{676}(329,\cdot)\)
\(\chi_{676}(381,\cdot)\)
\(\chi_{676}(413,\cdot)\)
\(\chi_{676}(433,\cdot)\)
\(\chi_{676}(465,\cdot)\)
\(\chi_{676}(517,\cdot)\)
\(\chi_{676}(537,\cdot)\)
\(\chi_{676}(569,\cdot)\)
\(\chi_{676}(589,\cdot)\)
\(\chi_{676}(621,\cdot)\)
\(\chi_{676}(641,\cdot)\)
\(\chi_{676}(673,\cdot)\)
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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 ]
\((339,509)\) → \((1,e\left(\frac{59}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 676 }(309, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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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)
