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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9386, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([228,92]))
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pari:[g,chi] = znchar(Mod(61,9386))
\(\chi_{9386}(9,\cdot)\)
\(\chi_{9386}(55,\cdot)\)
\(\chi_{9386}(61,\cdot)\)
\(\chi_{9386}(81,\cdot)\)
\(\chi_{9386}(139,\cdot)\)
\(\chi_{9386}(263,\cdot)\)
\(\chi_{9386}(503,\cdot)\)
\(\chi_{9386}(549,\cdot)\)
\(\chi_{9386}(555,\cdot)\)
\(\chi_{9386}(575,\cdot)\)
\(\chi_{9386}(633,\cdot)\)
\(\chi_{9386}(757,\cdot)\)
\(\chi_{9386}(997,\cdot)\)
\(\chi_{9386}(1043,\cdot)\)
\(\chi_{9386}(1049,\cdot)\)
\(\chi_{9386}(1069,\cdot)\)
\(\chi_{9386}(1127,\cdot)\)
\(\chi_{9386}(1251,\cdot)\)
\(\chi_{9386}(1491,\cdot)\)
\(\chi_{9386}(1537,\cdot)\)
\(\chi_{9386}(1563,\cdot)\)
\(\chi_{9386}(1621,\cdot)\)
\(\chi_{9386}(1745,\cdot)\)
\(\chi_{9386}(1985,\cdot)\)
\(\chi_{9386}(2031,\cdot)\)
\(\chi_{9386}(2037,\cdot)\)
\(\chi_{9386}(2057,\cdot)\)
\(\chi_{9386}(2115,\cdot)\)
\(\chi_{9386}(2239,\cdot)\)
\(\chi_{9386}(2479,\cdot)\)
...
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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 ]
\((1445,3251)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{46}{171}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9386 }(61, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{127}{171}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{140}{171}\right)\) |
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sage:chi.jacobi_sum(n)
