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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3042, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([39,37]))
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pari:[g,chi] = znchar(Mod(2123,3042))
\(\chi_{3042}(17,\cdot)\)
\(\chi_{3042}(179,\cdot)\)
\(\chi_{3042}(251,\cdot)\)
\(\chi_{3042}(413,\cdot)\)
\(\chi_{3042}(647,\cdot)\)
\(\chi_{3042}(719,\cdot)\)
\(\chi_{3042}(881,\cdot)\)
\(\chi_{3042}(953,\cdot)\)
\(\chi_{3042}(1115,\cdot)\)
\(\chi_{3042}(1187,\cdot)\)
\(\chi_{3042}(1349,\cdot)\)
\(\chi_{3042}(1421,\cdot)\)
\(\chi_{3042}(1583,\cdot)\)
\(\chi_{3042}(1655,\cdot)\)
\(\chi_{3042}(1817,\cdot)\)
\(\chi_{3042}(1889,\cdot)\)
\(\chi_{3042}(2123,\cdot)\)
\(\chi_{3042}(2285,\cdot)\)
\(\chi_{3042}(2357,\cdot)\)
\(\chi_{3042}(2519,\cdot)\)
\(\chi_{3042}(2591,\cdot)\)
\(\chi_{3042}(2753,\cdot)\)
\(\chi_{3042}(2825,\cdot)\)
\(\chi_{3042}(2987,\cdot)\)
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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 ]
\((677,847)\) → \((-1,e\left(\frac{37}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(2123, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{41}{78}\right)\) |
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sage:chi.jacobi_sum(n)
