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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,88,83]))
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pari:[g,chi] = znchar(Mod(3286,9075))
\(\chi_{9075}(46,\cdot)\)
\(\chi_{9075}(556,\cdot)\)
\(\chi_{9075}(811,\cdot)\)
\(\chi_{9075}(871,\cdot)\)
\(\chi_{9075}(1381,\cdot)\)
\(\chi_{9075}(1591,\cdot)\)
\(\chi_{9075}(1636,\cdot)\)
\(\chi_{9075}(1696,\cdot)\)
\(\chi_{9075}(2206,\cdot)\)
\(\chi_{9075}(2416,\cdot)\)
\(\chi_{9075}(2461,\cdot)\)
\(\chi_{9075}(2521,\cdot)\)
\(\chi_{9075}(3031,\cdot)\)
\(\chi_{9075}(3241,\cdot)\)
\(\chi_{9075}(3286,\cdot)\)
\(\chi_{9075}(3346,\cdot)\)
\(\chi_{9075}(3856,\cdot)\)
\(\chi_{9075}(4066,\cdot)\)
\(\chi_{9075}(4171,\cdot)\)
\(\chi_{9075}(4681,\cdot)\)
\(\chi_{9075}(4891,\cdot)\)
\(\chi_{9075}(4936,\cdot)\)
\(\chi_{9075}(4996,\cdot)\)
\(\chi_{9075}(5506,\cdot)\)
\(\chi_{9075}(5716,\cdot)\)
\(\chi_{9075}(5761,\cdot)\)
\(\chi_{9075}(5821,\cdot)\)
\(\chi_{9075}(6331,\cdot)\)
\(\chi_{9075}(6541,\cdot)\)
\(\chi_{9075}(6586,\cdot)\)
...
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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 ]
\((3026,727,5326)\) → \((1,e\left(\frac{4}{5}\right),e\left(\frac{83}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 9075 }(3286, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) |
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sage:chi.jacobi_sum(n)
