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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([0,11,378]))
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pari:[g,chi] = znchar(Mod(52,3381))
\(\chi_{3381}(52,\cdot)\)
\(\chi_{3381}(73,\cdot)\)
\(\chi_{3381}(82,\cdot)\)
\(\chi_{3381}(94,\cdot)\)
\(\chi_{3381}(124,\cdot)\)
\(\chi_{3381}(187,\cdot)\)
\(\chi_{3381}(220,\cdot)\)
\(\chi_{3381}(262,\cdot)\)
\(\chi_{3381}(271,\cdot)\)
\(\chi_{3381}(292,\cdot)\)
\(\chi_{3381}(334,\cdot)\)
\(\chi_{3381}(376,\cdot)\)
\(\chi_{3381}(397,\cdot)\)
\(\chi_{3381}(409,\cdot)\)
\(\chi_{3381}(418,\cdot)\)
\(\chi_{3381}(430,\cdot)\)
\(\chi_{3381}(439,\cdot)\)
\(\chi_{3381}(514,\cdot)\)
\(\chi_{3381}(535,\cdot)\)
\(\chi_{3381}(556,\cdot)\)
\(\chi_{3381}(565,\cdot)\)
\(\chi_{3381}(577,\cdot)\)
\(\chi_{3381}(670,\cdot)\)
\(\chi_{3381}(703,\cdot)\)
\(\chi_{3381}(745,\cdot)\)
\(\chi_{3381}(775,\cdot)\)
\(\chi_{3381}(808,\cdot)\)
\(\chi_{3381}(817,\cdot)\)
\(\chi_{3381}(859,\cdot)\)
\(\chi_{3381}(880,\cdot)\)
...
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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 ]
\((2255,346,442)\) → \((1,e\left(\frac{1}{42}\right),e\left(\frac{9}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3381 }(52, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{59}{231}\right)\) | \(e\left(\frac{118}{231}\right)\) | \(e\left(\frac{235}{462}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{353}{462}\right)\) | \(e\left(\frac{73}{231}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{149}{462}\right)\) | \(e\left(\frac{7}{66}\right)\) |
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sage:chi.jacobi_sum(n)
