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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3648, base_ring=CyclotomicField(72))
M = H._module
chi = DirichletCharacter(H, M([36,63,36,68]))
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pari:[g,chi] = znchar(Mod(599,3648))
\(\chi_{3648}(71,\cdot)\)
\(\chi_{3648}(167,\cdot)\)
\(\chi_{3648}(599,\cdot)\)
\(\chi_{3648}(743,\cdot)\)
\(\chi_{3648}(839,\cdot)\)
\(\chi_{3648}(887,\cdot)\)
\(\chi_{3648}(983,\cdot)\)
\(\chi_{3648}(1079,\cdot)\)
\(\chi_{3648}(1511,\cdot)\)
\(\chi_{3648}(1655,\cdot)\)
\(\chi_{3648}(1751,\cdot)\)
\(\chi_{3648}(1799,\cdot)\)
\(\chi_{3648}(1895,\cdot)\)
\(\chi_{3648}(1991,\cdot)\)
\(\chi_{3648}(2423,\cdot)\)
\(\chi_{3648}(2567,\cdot)\)
\(\chi_{3648}(2663,\cdot)\)
\(\chi_{3648}(2711,\cdot)\)
\(\chi_{3648}(2807,\cdot)\)
\(\chi_{3648}(2903,\cdot)\)
\(\chi_{3648}(3335,\cdot)\)
\(\chi_{3648}(3479,\cdot)\)
\(\chi_{3648}(3575,\cdot)\)
\(\chi_{3648}(3623,\cdot)\)
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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 ]
\((2623,2053,1217,1921)\) → \((-1,e\left(\frac{7}{8}\right),-1,e\left(\frac{17}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 3648 }(599, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{29}{72}\right)\) |
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sage:chi.jacobi_sum(n)
