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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([171,340]))
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pari:[g,chi] = znchar(Mod(1715,9386))
\(\chi_{9386}(25,\cdot)\)
\(\chi_{9386}(207,\cdot)\)
\(\chi_{9386}(233,\cdot)\)
\(\chi_{9386}(441,\cdot)\)
\(\chi_{9386}(519,\cdot)\)
\(\chi_{9386}(701,\cdot)\)
\(\chi_{9386}(727,\cdot)\)
\(\chi_{9386}(883,\cdot)\)
\(\chi_{9386}(909,\cdot)\)
\(\chi_{9386}(935,\cdot)\)
\(\chi_{9386}(1013,\cdot)\)
\(\chi_{9386}(1195,\cdot)\)
\(\chi_{9386}(1221,\cdot)\)
\(\chi_{9386}(1377,\cdot)\)
\(\chi_{9386}(1403,\cdot)\)
\(\chi_{9386}(1429,\cdot)\)
\(\chi_{9386}(1507,\cdot)\)
\(\chi_{9386}(1715,\cdot)\)
\(\chi_{9386}(1871,\cdot)\)
\(\chi_{9386}(1897,\cdot)\)
\(\chi_{9386}(1923,\cdot)\)
\(\chi_{9386}(2001,\cdot)\)
\(\chi_{9386}(2183,\cdot)\)
\(\chi_{9386}(2209,\cdot)\)
\(\chi_{9386}(2365,\cdot)\)
\(\chi_{9386}(2391,\cdot)\)
\(\chi_{9386}(2417,\cdot)\)
\(\chi_{9386}(2495,\cdot)\)
\(\chi_{9386}(2677,\cdot)\)
\(\chi_{9386}(2703,\cdot)\)
...
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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 ]
\((1445,3251)\) → \((-1,e\left(\frac{170}{171}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 9386 }(1715, a) \) |
\(1\) | \(1\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{49}{342}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{64}{171}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{113}{342}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{25}{171}\right)\) | \(e\left(\frac{49}{171}\right)\) |
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sage:chi.jacobi_sum(n)
