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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2667, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,63,94]))
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pari:[g,chi] = znchar(Mod(13,2667))
\(\chi_{2667}(13,\cdot)\)
\(\chi_{2667}(34,\cdot)\)
\(\chi_{2667}(265,\cdot)\)
\(\chi_{2667}(328,\cdot)\)
\(\chi_{2667}(412,\cdot)\)
\(\chi_{2667}(496,\cdot)\)
\(\chi_{2667}(517,\cdot)\)
\(\chi_{2667}(538,\cdot)\)
\(\chi_{2667}(580,\cdot)\)
\(\chi_{2667}(706,\cdot)\)
\(\chi_{2667}(748,\cdot)\)
\(\chi_{2667}(811,\cdot)\)
\(\chi_{2667}(832,\cdot)\)
\(\chi_{2667}(958,\cdot)\)
\(\chi_{2667}(1042,\cdot)\)
\(\chi_{2667}(1231,\cdot)\)
\(\chi_{2667}(1441,\cdot)\)
\(\chi_{2667}(1672,\cdot)\)
\(\chi_{2667}(1693,\cdot)\)
\(\chi_{2667}(1735,\cdot)\)
\(\chi_{2667}(1819,\cdot)\)
\(\chi_{2667}(1840,\cdot)\)
\(\chi_{2667}(1882,\cdot)\)
\(\chi_{2667}(1987,\cdot)\)
\(\chi_{2667}(2029,\cdot)\)
\(\chi_{2667}(2050,\cdot)\)
\(\chi_{2667}(2092,\cdot)\)
\(\chi_{2667}(2113,\cdot)\)
\(\chi_{2667}(2176,\cdot)\)
\(\chi_{2667}(2365,\cdot)\)
...
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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 ]
\((890,1144,2416)\) → \((1,-1,e\left(\frac{47}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2667 }(13, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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sage:chi.jacobi_sum(n)
