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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2646, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([119,45]))
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pari:[g,chi] = znchar(Mod(41,2646))
\(\chi_{2646}(41,\cdot)\)
\(\chi_{2646}(83,\cdot)\)
\(\chi_{2646}(167,\cdot)\)
\(\chi_{2646}(209,\cdot)\)
\(\chi_{2646}(335,\cdot)\)
\(\chi_{2646}(419,\cdot)\)
\(\chi_{2646}(461,\cdot)\)
\(\chi_{2646}(545,\cdot)\)
\(\chi_{2646}(671,\cdot)\)
\(\chi_{2646}(713,\cdot)\)
\(\chi_{2646}(797,\cdot)\)
\(\chi_{2646}(839,\cdot)\)
\(\chi_{2646}(923,\cdot)\)
\(\chi_{2646}(965,\cdot)\)
\(\chi_{2646}(1049,\cdot)\)
\(\chi_{2646}(1091,\cdot)\)
\(\chi_{2646}(1217,\cdot)\)
\(\chi_{2646}(1301,\cdot)\)
\(\chi_{2646}(1343,\cdot)\)
\(\chi_{2646}(1427,\cdot)\)
\(\chi_{2646}(1553,\cdot)\)
\(\chi_{2646}(1595,\cdot)\)
\(\chi_{2646}(1679,\cdot)\)
\(\chi_{2646}(1721,\cdot)\)
\(\chi_{2646}(1805,\cdot)\)
\(\chi_{2646}(1847,\cdot)\)
\(\chi_{2646}(1931,\cdot)\)
\(\chi_{2646}(1973,\cdot)\)
\(\chi_{2646}(2099,\cdot)\)
\(\chi_{2646}(2183,\cdot)\)
...
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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 ]
\((785,1081)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2646 }(41, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{21}\right)\) |
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sage:chi.jacobi_sum(n)
