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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([0,66,35]))
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pari:[g,chi] = znchar(Mod(1009,3381))
\(\chi_{3381}(43,\cdot)\)
\(\chi_{3381}(106,\cdot)\)
\(\chi_{3381}(274,\cdot)\)
\(\chi_{3381}(316,\cdot)\)
\(\chi_{3381}(337,\cdot)\)
\(\chi_{3381}(379,\cdot)\)
\(\chi_{3381}(421,\cdot)\)
\(\chi_{3381}(526,\cdot)\)
\(\chi_{3381}(631,\cdot)\)
\(\chi_{3381}(757,\cdot)\)
\(\chi_{3381}(778,\cdot)\)
\(\chi_{3381}(799,\cdot)\)
\(\chi_{3381}(820,\cdot)\)
\(\chi_{3381}(862,\cdot)\)
\(\chi_{3381}(904,\cdot)\)
\(\chi_{3381}(925,\cdot)\)
\(\chi_{3381}(1009,\cdot)\)
\(\chi_{3381}(1072,\cdot)\)
\(\chi_{3381}(1114,\cdot)\)
\(\chi_{3381}(1240,\cdot)\)
\(\chi_{3381}(1261,\cdot)\)
\(\chi_{3381}(1282,\cdot)\)
\(\chi_{3381}(1303,\cdot)\)
\(\chi_{3381}(1345,\cdot)\)
\(\chi_{3381}(1387,\cdot)\)
\(\chi_{3381}(1408,\cdot)\)
\(\chi_{3381}(1492,\cdot)\)
\(\chi_{3381}(1555,\cdot)\)
\(\chi_{3381}(1597,\cdot)\)
\(\chi_{3381}(1723,\cdot)\)
...
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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 ]
\((2255,346,442)\) → \((1,e\left(\frac{3}{7}\right),e\left(\frac{5}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3381 }(1009, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{101}{154}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{47}{154}\right)\) | \(e\left(\frac{9}{22}\right)\) |
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sage:chi.jacobi_sum(n)
