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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([54,0,10,81]))
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pari:[g,chi] = znchar(Mod(2543,3240))
\(\chi_{3240}(23,\cdot)\)
\(\chi_{3240}(47,\cdot)\)
\(\chi_{3240}(167,\cdot)\)
\(\chi_{3240}(263,\cdot)\)
\(\chi_{3240}(383,\cdot)\)
\(\chi_{3240}(407,\cdot)\)
\(\chi_{3240}(527,\cdot)\)
\(\chi_{3240}(623,\cdot)\)
\(\chi_{3240}(743,\cdot)\)
\(\chi_{3240}(767,\cdot)\)
\(\chi_{3240}(887,\cdot)\)
\(\chi_{3240}(983,\cdot)\)
\(\chi_{3240}(1103,\cdot)\)
\(\chi_{3240}(1127,\cdot)\)
\(\chi_{3240}(1247,\cdot)\)
\(\chi_{3240}(1343,\cdot)\)
\(\chi_{3240}(1463,\cdot)\)
\(\chi_{3240}(1487,\cdot)\)
\(\chi_{3240}(1607,\cdot)\)
\(\chi_{3240}(1703,\cdot)\)
\(\chi_{3240}(1823,\cdot)\)
\(\chi_{3240}(1847,\cdot)\)
\(\chi_{3240}(1967,\cdot)\)
\(\chi_{3240}(2063,\cdot)\)
\(\chi_{3240}(2183,\cdot)\)
\(\chi_{3240}(2207,\cdot)\)
\(\chi_{3240}(2327,\cdot)\)
\(\chi_{3240}(2423,\cdot)\)
\(\chi_{3240}(2543,\cdot)\)
\(\chi_{3240}(2567,\cdot)\)
...
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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 ]
\((2431,1621,3161,1297)\) → \((-1,1,e\left(\frac{5}{54}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(2543, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{49}{54}\right)\) |
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sage:chi.jacobi_sum(n)
