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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([0,58]))
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pari:[g,chi] = znchar(Mod(373,676))
\(\chi_{676}(9,\cdot)\)
\(\chi_{676}(29,\cdot)\)
\(\chi_{676}(61,\cdot)\)
\(\chi_{676}(81,\cdot)\)
\(\chi_{676}(113,\cdot)\)
\(\chi_{676}(133,\cdot)\)
\(\chi_{676}(165,\cdot)\)
\(\chi_{676}(185,\cdot)\)
\(\chi_{676}(217,\cdot)\)
\(\chi_{676}(237,\cdot)\)
\(\chi_{676}(269,\cdot)\)
\(\chi_{676}(289,\cdot)\)
\(\chi_{676}(321,\cdot)\)
\(\chi_{676}(341,\cdot)\)
\(\chi_{676}(373,\cdot)\)
\(\chi_{676}(393,\cdot)\)
\(\chi_{676}(425,\cdot)\)
\(\chi_{676}(445,\cdot)\)
\(\chi_{676}(477,\cdot)\)
\(\chi_{676}(497,\cdot)\)
\(\chi_{676}(549,\cdot)\)
\(\chi_{676}(581,\cdot)\)
\(\chi_{676}(601,\cdot)\)
\(\chi_{676}(633,\cdot)\)
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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 ]
\((339,509)\) → \((1,e\left(\frac{29}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 676 }(373, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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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)
