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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3645, base_ring=CyclotomicField(162))
M = H._module
chi = DirichletCharacter(H, M([86,0]))
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pari:[g,chi] = znchar(Mod(766,3645))
\(\chi_{3645}(46,\cdot)\)
\(\chi_{3645}(91,\cdot)\)
\(\chi_{3645}(181,\cdot)\)
\(\chi_{3645}(226,\cdot)\)
\(\chi_{3645}(316,\cdot)\)
\(\chi_{3645}(361,\cdot)\)
\(\chi_{3645}(451,\cdot)\)
\(\chi_{3645}(496,\cdot)\)
\(\chi_{3645}(586,\cdot)\)
\(\chi_{3645}(631,\cdot)\)
\(\chi_{3645}(721,\cdot)\)
\(\chi_{3645}(766,\cdot)\)
\(\chi_{3645}(856,\cdot)\)
\(\chi_{3645}(901,\cdot)\)
\(\chi_{3645}(991,\cdot)\)
\(\chi_{3645}(1036,\cdot)\)
\(\chi_{3645}(1126,\cdot)\)
\(\chi_{3645}(1171,\cdot)\)
\(\chi_{3645}(1261,\cdot)\)
\(\chi_{3645}(1306,\cdot)\)
\(\chi_{3645}(1396,\cdot)\)
\(\chi_{3645}(1441,\cdot)\)
\(\chi_{3645}(1531,\cdot)\)
\(\chi_{3645}(1576,\cdot)\)
\(\chi_{3645}(1666,\cdot)\)
\(\chi_{3645}(1711,\cdot)\)
\(\chi_{3645}(1801,\cdot)\)
\(\chi_{3645}(1846,\cdot)\)
\(\chi_{3645}(1936,\cdot)\)
\(\chi_{3645}(1981,\cdot)\)
...
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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 ]
\((731,2917)\) → \((e\left(\frac{43}{81}\right),1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3645 }(766, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{81}\right)\) | \(e\left(\frac{5}{81}\right)\) | \(e\left(\frac{13}{81}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{19}{81}\right)\) | \(e\left(\frac{20}{81}\right)\) | \(e\left(\frac{56}{81}\right)\) | \(e\left(\frac{10}{81}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) |
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sage:chi.jacobi_sum(n)
