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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([165,213,275]))
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pari:[g,chi] = znchar(Mod(530,14157))
\(\chi_{14157}(17,\cdot)\)
\(\chi_{14157}(62,\cdot)\)
\(\chi_{14157}(134,\cdot)\)
\(\chi_{14157}(413,\cdot)\)
\(\chi_{14157}(530,\cdot)\)
\(\chi_{14157}(953,\cdot)\)
\(\chi_{14157}(998,\cdot)\)
\(\chi_{14157}(1349,\cdot)\)
\(\chi_{14157}(1421,\cdot)\)
\(\chi_{14157}(1700,\cdot)\)
\(\chi_{14157}(1817,\cdot)\)
\(\chi_{14157}(1889,\cdot)\)
\(\chi_{14157}(2240,\cdot)\)
\(\chi_{14157}(2285,\cdot)\)
\(\chi_{14157}(2591,\cdot)\)
\(\chi_{14157}(2636,\cdot)\)
\(\chi_{14157}(2708,\cdot)\)
\(\chi_{14157}(2987,\cdot)\)
\(\chi_{14157}(3104,\cdot)\)
\(\chi_{14157}(3176,\cdot)\)
\(\chi_{14157}(3527,\cdot)\)
\(\chi_{14157}(3572,\cdot)\)
\(\chi_{14157}(3878,\cdot)\)
\(\chi_{14157}(3923,\cdot)\)
\(\chi_{14157}(3995,\cdot)\)
\(\chi_{14157}(4274,\cdot)\)
\(\chi_{14157}(4391,\cdot)\)
\(\chi_{14157}(4463,\cdot)\)
\(\chi_{14157}(4814,\cdot)\)
\(\chi_{14157}(4859,\cdot)\)
...
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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 ]
\((6293,3511,4357)\) → \((-1,e\left(\frac{71}{110}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 14157 }(530, a) \) |
\(1\) | \(1\) | \(e\left(\frac{323}{330}\right)\) | \(e\left(\frac{158}{165}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{113}{165}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{151}{165}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{122}{165}\right)\) |
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sage:chi.jacobi_sum(n)
