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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9386, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([21,26]))
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pari:[g,chi] = znchar(Mod(4639,9386))
\(\chi_{9386}(1029,\cdot)\)
\(\chi_{9386}(1345,\cdot)\)
\(\chi_{9386}(2789,\cdot)\)
\(\chi_{9386}(3187,\cdot)\)
\(\chi_{9386}(3365,\cdot)\)
\(\chi_{9386}(3737,\cdot)\)
\(\chi_{9386}(4639,\cdot)\)
\(\chi_{9386}(5181,\cdot)\)
\(\chi_{9386}(6797,\cdot)\)
\(\chi_{9386}(6831,\cdot)\)
\(\chi_{9386}(8275,\cdot)\)
\(\chi_{9386}(9141,\cdot)\)
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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 ]
\((1445,3251)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{13}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9386 }(4639, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(-i\) | \(e\left(\frac{4}{9}\right)\) | \(-i\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) |
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sage:chi.jacobi_sum(n)
