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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([52,49]))
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pari:[g,chi] = znchar(Mod(745,3042))
\(\chi_{3042}(43,\cdot)\)
\(\chi_{3042}(49,\cdot)\)
\(\chi_{3042}(277,\cdot)\)
\(\chi_{3042}(283,\cdot)\)
\(\chi_{3042}(511,\cdot)\)
\(\chi_{3042}(517,\cdot)\)
\(\chi_{3042}(745,\cdot)\)
\(\chi_{3042}(751,\cdot)\)
\(\chi_{3042}(979,\cdot)\)
\(\chi_{3042}(985,\cdot)\)
\(\chi_{3042}(1213,\cdot)\)
\(\chi_{3042}(1219,\cdot)\)
\(\chi_{3042}(1447,\cdot)\)
\(\chi_{3042}(1453,\cdot)\)
\(\chi_{3042}(1681,\cdot)\)
\(\chi_{3042}(1687,\cdot)\)
\(\chi_{3042}(1915,\cdot)\)
\(\chi_{3042}(1921,\cdot)\)
\(\chi_{3042}(2149,\cdot)\)
\(\chi_{3042}(2155,\cdot)\)
\(\chi_{3042}(2383,\cdot)\)
\(\chi_{3042}(2617,\cdot)\)
\(\chi_{3042}(2623,\cdot)\)
\(\chi_{3042}(2857,\cdot)\)
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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 ]
\((677,847)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{49}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(745, a) \) |
\(1\) | \(1\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{34}{39}\right)\) |
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sage:chi.jacobi_sum(n)
