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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4563, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([104,131]))
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pari:[g,chi] = znchar(Mod(46,4563))
\(\chi_{4563}(37,\cdot)\)
\(\chi_{4563}(46,\cdot)\)
\(\chi_{4563}(145,\cdot)\)
\(\chi_{4563}(370,\cdot)\)
\(\chi_{4563}(388,\cdot)\)
\(\chi_{4563}(397,\cdot)\)
\(\chi_{4563}(496,\cdot)\)
\(\chi_{4563}(721,\cdot)\)
\(\chi_{4563}(739,\cdot)\)
\(\chi_{4563}(748,\cdot)\)
\(\chi_{4563}(847,\cdot)\)
\(\chi_{4563}(1072,\cdot)\)
\(\chi_{4563}(1090,\cdot)\)
\(\chi_{4563}(1099,\cdot)\)
\(\chi_{4563}(1198,\cdot)\)
\(\chi_{4563}(1423,\cdot)\)
\(\chi_{4563}(1450,\cdot)\)
\(\chi_{4563}(1549,\cdot)\)
\(\chi_{4563}(1774,\cdot)\)
\(\chi_{4563}(1792,\cdot)\)
\(\chi_{4563}(1801,\cdot)\)
\(\chi_{4563}(1900,\cdot)\)
\(\chi_{4563}(2125,\cdot)\)
\(\chi_{4563}(2143,\cdot)\)
\(\chi_{4563}(2152,\cdot)\)
\(\chi_{4563}(2251,\cdot)\)
\(\chi_{4563}(2476,\cdot)\)
\(\chi_{4563}(2494,\cdot)\)
\(\chi_{4563}(2503,\cdot)\)
\(\chi_{4563}(2602,\cdot)\)
...
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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 ]
\((677,3889)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{131}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4563 }(46, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{47}{78}\right)\) |
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sage:chi.jacobi_sum(n)
