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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(254144, base_ring=CyclotomicField(760))
M = H._module
chi = DirichletCharacter(H, M([380,285,76,120]))
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pari:[g,chi] = znchar(Mod(7031,254144))
\(\chi_{254144}(39,\cdot)\)
\(\chi_{254144}(343,\cdot)\)
\(\chi_{254144}(1559,\cdot)\)
\(\chi_{254144}(2471,\cdot)\)
\(\chi_{254144}(3383,\cdot)\)
\(\chi_{254144}(3687,\cdot)\)
\(\chi_{254144}(4903,\cdot)\)
\(\chi_{254144}(5815,\cdot)\)
\(\chi_{254144}(6727,\cdot)\)
\(\chi_{254144}(7031,\cdot)\)
\(\chi_{254144}(8247,\cdot)\)
\(\chi_{254144}(9159,\cdot)\)
\(\chi_{254144}(10071,\cdot)\)
\(\chi_{254144}(10375,\cdot)\)
\(\chi_{254144}(11591,\cdot)\)
\(\chi_{254144}(12503,\cdot)\)
\(\chi_{254144}(13415,\cdot)\)
\(\chi_{254144}(14935,\cdot)\)
\(\chi_{254144}(15847,\cdot)\)
\(\chi_{254144}(16759,\cdot)\)
\(\chi_{254144}(17063,\cdot)\)
\(\chi_{254144}(18279,\cdot)\)
\(\chi_{254144}(19191,\cdot)\)
\(\chi_{254144}(20103,\cdot)\)
\(\chi_{254144}(20407,\cdot)\)
\(\chi_{254144}(21623,\cdot)\)
\(\chi_{254144}(22535,\cdot)\)
\(\chi_{254144}(23447,\cdot)\)
\(\chi_{254144}(23751,\cdot)\)
\(\chi_{254144}(24967,\cdot)\)
...
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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 ]
\((166783,174725,69313,14081)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{1}{10}\right),e\left(\frac{3}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 254144 }(7031, a) \) |
\(1\) | \(1\) | \(e\left(\frac{283}{760}\right)\) | \(e\left(\frac{309}{760}\right)\) | \(e\left(\frac{241}{380}\right)\) | \(e\left(\frac{283}{380}\right)\) | \(e\left(\frac{511}{760}\right)\) | \(e\left(\frac{74}{95}\right)\) | \(e\left(\frac{8}{95}\right)\) | \(e\left(\frac{1}{152}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{309}{380}\right)\) |
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sage:chi.jacobi_sum(n)
