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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([130,85]))
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pari:[g,chi] = znchar(Mod(41,3042))
\(\chi_{3042}(41,\cdot)\)
\(\chi_{3042}(137,\cdot)\)
\(\chi_{3042}(167,\cdot)\)
\(\chi_{3042}(227,\cdot)\)
\(\chi_{3042}(275,\cdot)\)
\(\chi_{3042}(371,\cdot)\)
\(\chi_{3042}(401,\cdot)\)
\(\chi_{3042}(461,\cdot)\)
\(\chi_{3042}(509,\cdot)\)
\(\chi_{3042}(605,\cdot)\)
\(\chi_{3042}(635,\cdot)\)
\(\chi_{3042}(743,\cdot)\)
\(\chi_{3042}(839,\cdot)\)
\(\chi_{3042}(869,\cdot)\)
\(\chi_{3042}(929,\cdot)\)
\(\chi_{3042}(977,\cdot)\)
\(\chi_{3042}(1073,\cdot)\)
\(\chi_{3042}(1163,\cdot)\)
\(\chi_{3042}(1211,\cdot)\)
\(\chi_{3042}(1307,\cdot)\)
\(\chi_{3042}(1337,\cdot)\)
\(\chi_{3042}(1397,\cdot)\)
\(\chi_{3042}(1445,\cdot)\)
\(\chi_{3042}(1541,\cdot)\)
\(\chi_{3042}(1571,\cdot)\)
\(\chi_{3042}(1631,\cdot)\)
\(\chi_{3042}(1679,\cdot)\)
\(\chi_{3042}(1775,\cdot)\)
\(\chi_{3042}(1805,\cdot)\)
\(\chi_{3042}(1865,\cdot)\)
...
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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 ]
\((677,847)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{85}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(41, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{55}{78}\right)\) |
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sage:chi.jacobi_sum(n)
