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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14157, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([165,189,110]))
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pari:[g,chi] = znchar(Mod(107,14157))
\(\chi_{14157}(35,\cdot)\)
\(\chi_{14157}(107,\cdot)\)
\(\chi_{14157}(458,\cdot)\)
\(\chi_{14157}(503,\cdot)\)
\(\chi_{14157}(809,\cdot)\)
\(\chi_{14157}(854,\cdot)\)
\(\chi_{14157}(926,\cdot)\)
\(\chi_{14157}(1205,\cdot)\)
\(\chi_{14157}(1394,\cdot)\)
\(\chi_{14157}(1745,\cdot)\)
\(\chi_{14157}(1790,\cdot)\)
\(\chi_{14157}(2096,\cdot)\)
\(\chi_{14157}(2141,\cdot)\)
\(\chi_{14157}(2213,\cdot)\)
\(\chi_{14157}(2492,\cdot)\)
\(\chi_{14157}(2609,\cdot)\)
\(\chi_{14157}(2681,\cdot)\)
\(\chi_{14157}(3032,\cdot)\)
\(\chi_{14157}(3077,\cdot)\)
\(\chi_{14157}(3383,\cdot)\)
\(\chi_{14157}(3779,\cdot)\)
\(\chi_{14157}(3896,\cdot)\)
\(\chi_{14157}(3968,\cdot)\)
\(\chi_{14157}(4319,\cdot)\)
\(\chi_{14157}(4364,\cdot)\)
\(\chi_{14157}(4670,\cdot)\)
\(\chi_{14157}(4715,\cdot)\)
\(\chi_{14157}(4787,\cdot)\)
\(\chi_{14157}(5066,\cdot)\)
\(\chi_{14157}(5183,\cdot)\)
...
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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 ]
\((6293,3511,4357)\) → \((-1,e\left(\frac{63}{110}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 14157 }(107, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{165}\right)\) | \(e\left(\frac{134}{165}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{223}{330}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{103}{165}\right)\) | \(e\left(\frac{38}{165}\right)\) | \(e\left(\frac{67}{330}\right)\) |
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sage:chi.jacobi_sum(n)
