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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3042, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([104,107]))
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pari:[g,chi] = znchar(Mod(7,3042))
\(\chi_{3042}(7,\cdot)\)
\(\chi_{3042}(67,\cdot)\)
\(\chi_{3042}(97,\cdot)\)
\(\chi_{3042}(193,\cdot)\)
\(\chi_{3042}(241,\cdot)\)
\(\chi_{3042}(301,\cdot)\)
\(\chi_{3042}(331,\cdot)\)
\(\chi_{3042}(475,\cdot)\)
\(\chi_{3042}(535,\cdot)\)
\(\chi_{3042}(565,\cdot)\)
\(\chi_{3042}(661,\cdot)\)
\(\chi_{3042}(709,\cdot)\)
\(\chi_{3042}(769,\cdot)\)
\(\chi_{3042}(799,\cdot)\)
\(\chi_{3042}(895,\cdot)\)
\(\chi_{3042}(943,\cdot)\)
\(\chi_{3042}(1003,\cdot)\)
\(\chi_{3042}(1129,\cdot)\)
\(\chi_{3042}(1177,\cdot)\)
\(\chi_{3042}(1237,\cdot)\)
\(\chi_{3042}(1267,\cdot)\)
\(\chi_{3042}(1363,\cdot)\)
\(\chi_{3042}(1411,\cdot)\)
\(\chi_{3042}(1471,\cdot)\)
\(\chi_{3042}(1501,\cdot)\)
\(\chi_{3042}(1597,\cdot)\)
\(\chi_{3042}(1645,\cdot)\)
\(\chi_{3042}(1705,\cdot)\)
\(\chi_{3042}(1735,\cdot)\)
\(\chi_{3042}(1831,\cdot)\)
...
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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 ]
\((677,847)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{107}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(7, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{22}{39}\right)\) |
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sage:chi.jacobi_sum(n)
