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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2900, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([0,91,75]))
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pari:[g,chi] = znchar(Mod(317,2900))
\(\chi_{2900}(37,\cdot)\)
\(\chi_{2900}(97,\cdot)\)
\(\chi_{2900}(253,\cdot)\)
\(\chi_{2900}(317,\cdot)\)
\(\chi_{2900}(333,\cdot)\)
\(\chi_{2900}(337,\cdot)\)
\(\chi_{2900}(533,\cdot)\)
\(\chi_{2900}(537,\cdot)\)
\(\chi_{2900}(553,\cdot)\)
\(\chi_{2900}(577,\cdot)\)
\(\chi_{2900}(617,\cdot)\)
\(\chi_{2900}(677,\cdot)\)
\(\chi_{2900}(773,\cdot)\)
\(\chi_{2900}(833,\cdot)\)
\(\chi_{2900}(873,\cdot)\)
\(\chi_{2900}(897,\cdot)\)
\(\chi_{2900}(913,\cdot)\)
\(\chi_{2900}(917,\cdot)\)
\(\chi_{2900}(1113,\cdot)\)
\(\chi_{2900}(1117,\cdot)\)
\(\chi_{2900}(1133,\cdot)\)
\(\chi_{2900}(1197,\cdot)\)
\(\chi_{2900}(1353,\cdot)\)
\(\chi_{2900}(1413,\cdot)\)
\(\chi_{2900}(1453,\cdot)\)
\(\chi_{2900}(1477,\cdot)\)
\(\chi_{2900}(1497,\cdot)\)
\(\chi_{2900}(1697,\cdot)\)
\(\chi_{2900}(1713,\cdot)\)
\(\chi_{2900}(1737,\cdot)\)
...
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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 ]
\((1451,1277,901)\) → \((1,e\left(\frac{13}{20}\right),e\left(\frac{15}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2900 }(317, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{24}{35}\right)\) |
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sage:chi.jacobi_sum(n)
