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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3762, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([0,72,70]))
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pari:[g,chi] = znchar(Mod(1279,3762))
\(\chi_{3762}(289,\cdot)\)
\(\chi_{3762}(631,\cdot)\)
\(\chi_{3762}(757,\cdot)\)
\(\chi_{3762}(883,\cdot)\)
\(\chi_{3762}(955,\cdot)\)
\(\chi_{3762}(973,\cdot)\)
\(\chi_{3762}(1081,\cdot)\)
\(\chi_{3762}(1225,\cdot)\)
\(\chi_{3762}(1279,\cdot)\)
\(\chi_{3762}(1423,\cdot)\)
\(\chi_{3762}(1567,\cdot)\)
\(\chi_{3762}(1621,\cdot)\)
\(\chi_{3762}(1765,\cdot)\)
\(\chi_{3762}(1963,\cdot)\)
\(\chi_{3762}(2341,\cdot)\)
\(\chi_{3762}(2467,\cdot)\)
\(\chi_{3762}(2665,\cdot)\)
\(\chi_{3762}(2809,\cdot)\)
\(\chi_{3762}(2935,\cdot)\)
\(\chi_{3762}(3007,\cdot)\)
\(\chi_{3762}(3133,\cdot)\)
\(\chi_{3762}(3151,\cdot)\)
\(\chi_{3762}(3331,\cdot)\)
\(\chi_{3762}(3349,\cdot)\)
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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 ]
\((2927,343,2377)\) → \((1,e\left(\frac{4}{5}\right),e\left(\frac{7}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 3762 }(1279, a) \) |
\(1\) | \(1\) | \(e\left(\frac{29}{45}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{44}{45}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{45}\right)\) | \(e\left(\frac{37}{45}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{45}\right)\) | \(e\left(\frac{3}{5}\right)\) |
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sage:chi.jacobi_sum(n)
