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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3475, base_ring=CyclotomicField(276))
M = H._module
chi = DirichletCharacter(H, M([69,178]))
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pari:[g,chi] = znchar(Mod(457,3475))
\(\chi_{3475}(18,\cdot)\)
\(\chi_{3475}(32,\cdot)\)
\(\chi_{3475}(68,\cdot)\)
\(\chi_{3475}(93,\cdot)\)
\(\chi_{3475}(132,\cdot)\)
\(\chi_{3475}(157,\cdot)\)
\(\chi_{3475}(207,\cdot)\)
\(\chi_{3475}(232,\cdot)\)
\(\chi_{3475}(243,\cdot)\)
\(\chi_{3475}(293,\cdot)\)
\(\chi_{3475}(318,\cdot)\)
\(\chi_{3475}(368,\cdot)\)
\(\chi_{3475}(382,\cdot)\)
\(\chi_{3475}(393,\cdot)\)
\(\chi_{3475}(432,\cdot)\)
\(\chi_{3475}(443,\cdot)\)
\(\chi_{3475}(457,\cdot)\)
\(\chi_{3475}(507,\cdot)\)
\(\chi_{3475}(518,\cdot)\)
\(\chi_{3475}(532,\cdot)\)
\(\chi_{3475}(543,\cdot)\)
\(\chi_{3475}(568,\cdot)\)
\(\chi_{3475}(582,\cdot)\)
\(\chi_{3475}(657,\cdot)\)
\(\chi_{3475}(682,\cdot)\)
\(\chi_{3475}(707,\cdot)\)
\(\chi_{3475}(768,\cdot)\)
\(\chi_{3475}(793,\cdot)\)
\(\chi_{3475}(818,\cdot)\)
\(\chi_{3475}(907,\cdot)\)
...
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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 ]
\((1252,2226)\) → \((i,e\left(\frac{89}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3475 }(457, a) \) |
\(1\) | \(1\) | \(e\left(\frac{247}{276}\right)\) | \(e\left(\frac{53}{276}\right)\) | \(e\left(\frac{109}{138}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{137}{276}\right)\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{53}{138}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{271}{276}\right)\) | \(e\left(\frac{7}{276}\right)\) |
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sage:chi.jacobi_sum(n)
