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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([0,0,88,27]))
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pari:[g,chi] = znchar(Mod(2497,3240))
\(\chi_{3240}(97,\cdot)\)
\(\chi_{3240}(193,\cdot)\)
\(\chi_{3240}(313,\cdot)\)
\(\chi_{3240}(337,\cdot)\)
\(\chi_{3240}(457,\cdot)\)
\(\chi_{3240}(553,\cdot)\)
\(\chi_{3240}(673,\cdot)\)
\(\chi_{3240}(697,\cdot)\)
\(\chi_{3240}(817,\cdot)\)
\(\chi_{3240}(913,\cdot)\)
\(\chi_{3240}(1033,\cdot)\)
\(\chi_{3240}(1057,\cdot)\)
\(\chi_{3240}(1177,\cdot)\)
\(\chi_{3240}(1273,\cdot)\)
\(\chi_{3240}(1393,\cdot)\)
\(\chi_{3240}(1417,\cdot)\)
\(\chi_{3240}(1537,\cdot)\)
\(\chi_{3240}(1633,\cdot)\)
\(\chi_{3240}(1753,\cdot)\)
\(\chi_{3240}(1777,\cdot)\)
\(\chi_{3240}(1897,\cdot)\)
\(\chi_{3240}(1993,\cdot)\)
\(\chi_{3240}(2113,\cdot)\)
\(\chi_{3240}(2137,\cdot)\)
\(\chi_{3240}(2257,\cdot)\)
\(\chi_{3240}(2353,\cdot)\)
\(\chi_{3240}(2473,\cdot)\)
\(\chi_{3240}(2497,\cdot)\)
\(\chi_{3240}(2617,\cdot)\)
\(\chi_{3240}(2713,\cdot)\)
...
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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 ]
\((2431,1621,3161,1297)\) → \((1,1,e\left(\frac{22}{27}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(2497, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{5}{27}\right)\) |
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sage:chi.jacobi_sum(n)
