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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43904, base_ring=CyclotomicField(784))
M = H._module
chi = DirichletCharacter(H, M([0,147,24]))
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pari:[g,chi] = znchar(Mod(1161,43904))
\(\chi_{43904}(41,\cdot)\)
\(\chi_{43904}(153,\cdot)\)
\(\chi_{43904}(265,\cdot)\)
\(\chi_{43904}(377,\cdot)\)
\(\chi_{43904}(601,\cdot)\)
\(\chi_{43904}(713,\cdot)\)
\(\chi_{43904}(825,\cdot)\)
\(\chi_{43904}(937,\cdot)\)
\(\chi_{43904}(1049,\cdot)\)
\(\chi_{43904}(1161,\cdot)\)
\(\chi_{43904}(1385,\cdot)\)
\(\chi_{43904}(1497,\cdot)\)
\(\chi_{43904}(1609,\cdot)\)
\(\chi_{43904}(1721,\cdot)\)
\(\chi_{43904}(1833,\cdot)\)
\(\chi_{43904}(1945,\cdot)\)
\(\chi_{43904}(2169,\cdot)\)
\(\chi_{43904}(2281,\cdot)\)
\(\chi_{43904}(2393,\cdot)\)
\(\chi_{43904}(2505,\cdot)\)
\(\chi_{43904}(2617,\cdot)\)
\(\chi_{43904}(2729,\cdot)\)
\(\chi_{43904}(2953,\cdot)\)
\(\chi_{43904}(3065,\cdot)\)
\(\chi_{43904}(3177,\cdot)\)
\(\chi_{43904}(3289,\cdot)\)
\(\chi_{43904}(3401,\cdot)\)
\(\chi_{43904}(3513,\cdot)\)
\(\chi_{43904}(3737,\cdot)\)
\(\chi_{43904}(3849,\cdot)\)
...
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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 ]
\((17151,9605,17153)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{3}{98}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 43904 }(1161, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{465}{784}\right)\) | \(e\left(\frac{59}{784}\right)\) | \(e\left(\frac{73}{392}\right)\) | \(e\left(\frac{239}{784}\right)\) | \(e\left(\frac{421}{784}\right)\) | \(e\left(\frac{131}{196}\right)\) | \(e\left(\frac{3}{196}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{253}{392}\right)\) | \(e\left(\frac{59}{392}\right)\) |
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sage:chi.jacobi_sum(n)
