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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([0,11,2]))
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pari:[g,chi] = znchar(Mod(4,9075))
\(\chi_{9075}(4,\cdot)\)
\(\chi_{9075}(64,\cdot)\)
\(\chi_{9075}(619,\cdot)\)
\(\chi_{9075}(709,\cdot)\)
\(\chi_{9075}(829,\cdot)\)
\(\chi_{9075}(889,\cdot)\)
\(\chi_{9075}(1444,\cdot)\)
\(\chi_{9075}(1534,\cdot)\)
\(\chi_{9075}(1714,\cdot)\)
\(\chi_{9075}(2269,\cdot)\)
\(\chi_{9075}(2359,\cdot)\)
\(\chi_{9075}(2479,\cdot)\)
\(\chi_{9075}(2539,\cdot)\)
\(\chi_{9075}(3094,\cdot)\)
\(\chi_{9075}(3184,\cdot)\)
\(\chi_{9075}(3304,\cdot)\)
\(\chi_{9075}(3364,\cdot)\)
\(\chi_{9075}(3919,\cdot)\)
\(\chi_{9075}(4009,\cdot)\)
\(\chi_{9075}(4129,\cdot)\)
\(\chi_{9075}(4189,\cdot)\)
\(\chi_{9075}(4744,\cdot)\)
\(\chi_{9075}(4834,\cdot)\)
\(\chi_{9075}(4954,\cdot)\)
\(\chi_{9075}(5014,\cdot)\)
\(\chi_{9075}(5659,\cdot)\)
\(\chi_{9075}(5779,\cdot)\)
\(\chi_{9075}(5839,\cdot)\)
\(\chi_{9075}(6394,\cdot)\)
\(\chi_{9075}(6484,\cdot)\)
...
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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 ]
\((3026,727,5326)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{1}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 9075 }(4, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) |
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sage:chi.jacobi_sum(n)
