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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2349, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([35,99]))
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pari:[g,chi] = znchar(Mod(1745,2349))
\(\chi_{2349}(35,\cdot)\)
\(\chi_{2349}(62,\cdot)\)
\(\chi_{2349}(71,\cdot)\)
\(\chi_{2349}(125,\cdot)\)
\(\chi_{2349}(179,\cdot)\)
\(\chi_{2349}(332,\cdot)\)
\(\chi_{2349}(341,\cdot)\)
\(\chi_{2349}(386,\cdot)\)
\(\chi_{2349}(440,\cdot)\)
\(\chi_{2349}(557,\cdot)\)
\(\chi_{2349}(584,\cdot)\)
\(\chi_{2349}(602,\cdot)\)
\(\chi_{2349}(818,\cdot)\)
\(\chi_{2349}(845,\cdot)\)
\(\chi_{2349}(854,\cdot)\)
\(\chi_{2349}(908,\cdot)\)
\(\chi_{2349}(962,\cdot)\)
\(\chi_{2349}(1115,\cdot)\)
\(\chi_{2349}(1124,\cdot)\)
\(\chi_{2349}(1169,\cdot)\)
\(\chi_{2349}(1223,\cdot)\)
\(\chi_{2349}(1340,\cdot)\)
\(\chi_{2349}(1367,\cdot)\)
\(\chi_{2349}(1385,\cdot)\)
\(\chi_{2349}(1601,\cdot)\)
\(\chi_{2349}(1628,\cdot)\)
\(\chi_{2349}(1637,\cdot)\)
\(\chi_{2349}(1691,\cdot)\)
\(\chi_{2349}(1745,\cdot)\)
\(\chi_{2349}(1898,\cdot)\)
...
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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 ]
\((407,1945)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2349 }(1745, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) |
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sage:chi.jacobi_sum(n)
