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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([65,27]))
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pari:[g,chi] = znchar(Mod(77,3042))
\(\chi_{3042}(77,\cdot)\)
\(\chi_{3042}(155,\cdot)\)
\(\chi_{3042}(311,\cdot)\)
\(\chi_{3042}(389,\cdot)\)
\(\chi_{3042}(545,\cdot)\)
\(\chi_{3042}(623,\cdot)\)
\(\chi_{3042}(779,\cdot)\)
\(\chi_{3042}(857,\cdot)\)
\(\chi_{3042}(1091,\cdot)\)
\(\chi_{3042}(1247,\cdot)\)
\(\chi_{3042}(1325,\cdot)\)
\(\chi_{3042}(1481,\cdot)\)
\(\chi_{3042}(1559,\cdot)\)
\(\chi_{3042}(1715,\cdot)\)
\(\chi_{3042}(1793,\cdot)\)
\(\chi_{3042}(1949,\cdot)\)
\(\chi_{3042}(2183,\cdot)\)
\(\chi_{3042}(2261,\cdot)\)
\(\chi_{3042}(2417,\cdot)\)
\(\chi_{3042}(2495,\cdot)\)
\(\chi_{3042}(2651,\cdot)\)
\(\chi_{3042}(2729,\cdot)\)
\(\chi_{3042}(2885,\cdot)\)
\(\chi_{3042}(2963,\cdot)\)
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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 ]
\((677,847)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(77, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{17}{26}\right)\) |
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sage:chi.jacobi_sum(n)
