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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([0,151]))
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pari:[g,chi] = znchar(Mod(37,3042))
\(\chi_{3042}(37,\cdot)\)
\(\chi_{3042}(145,\cdot)\)
\(\chi_{3042}(163,\cdot)\)
\(\chi_{3042}(253,\cdot)\)
\(\chi_{3042}(271,\cdot)\)
\(\chi_{3042}(379,\cdot)\)
\(\chi_{3042}(397,\cdot)\)
\(\chi_{3042}(487,\cdot)\)
\(\chi_{3042}(505,\cdot)\)
\(\chi_{3042}(613,\cdot)\)
\(\chi_{3042}(631,\cdot)\)
\(\chi_{3042}(721,\cdot)\)
\(\chi_{3042}(739,\cdot)\)
\(\chi_{3042}(847,\cdot)\)
\(\chi_{3042}(865,\cdot)\)
\(\chi_{3042}(955,\cdot)\)
\(\chi_{3042}(973,\cdot)\)
\(\chi_{3042}(1081,\cdot)\)
\(\chi_{3042}(1099,\cdot)\)
\(\chi_{3042}(1189,\cdot)\)
\(\chi_{3042}(1207,\cdot)\)
\(\chi_{3042}(1315,\cdot)\)
\(\chi_{3042}(1423,\cdot)\)
\(\chi_{3042}(1549,\cdot)\)
\(\chi_{3042}(1567,\cdot)\)
\(\chi_{3042}(1657,\cdot)\)
\(\chi_{3042}(1675,\cdot)\)
\(\chi_{3042}(1783,\cdot)\)
\(\chi_{3042}(1801,\cdot)\)
\(\chi_{3042}(1891,\cdot)\)
...
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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 ]
\((677,847)\) → \((1,e\left(\frac{151}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3042 }(37, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{11}{39}\right)\) |
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sage:chi.jacobi_sum(n)
