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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4864, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([72,27,16]))
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pari:[g,chi] = znchar(Mod(175,4864))
\(\chi_{4864}(47,\cdot)\)
\(\chi_{4864}(111,\cdot)\)
\(\chi_{4864}(175,\cdot)\)
\(\chi_{4864}(207,\cdot)\)
\(\chi_{4864}(271,\cdot)\)
\(\chi_{4864}(367,\cdot)\)
\(\chi_{4864}(655,\cdot)\)
\(\chi_{4864}(719,\cdot)\)
\(\chi_{4864}(783,\cdot)\)
\(\chi_{4864}(815,\cdot)\)
\(\chi_{4864}(879,\cdot)\)
\(\chi_{4864}(975,\cdot)\)
\(\chi_{4864}(1263,\cdot)\)
\(\chi_{4864}(1327,\cdot)\)
\(\chi_{4864}(1391,\cdot)\)
\(\chi_{4864}(1423,\cdot)\)
\(\chi_{4864}(1487,\cdot)\)
\(\chi_{4864}(1583,\cdot)\)
\(\chi_{4864}(1871,\cdot)\)
\(\chi_{4864}(1935,\cdot)\)
\(\chi_{4864}(1999,\cdot)\)
\(\chi_{4864}(2031,\cdot)\)
\(\chi_{4864}(2095,\cdot)\)
\(\chi_{4864}(2191,\cdot)\)
\(\chi_{4864}(2479,\cdot)\)
\(\chi_{4864}(2543,\cdot)\)
\(\chi_{4864}(2607,\cdot)\)
\(\chi_{4864}(2639,\cdot)\)
\(\chi_{4864}(2703,\cdot)\)
\(\chi_{4864}(2799,\cdot)\)
...
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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 ]
\((3839,2053,4353)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 4864 }(175, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{79}{144}\right)\) | \(e\left(\frac{25}{72}\right)\) |
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sage:chi.jacobi_sum(n)
