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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4563, base_ring=CyclotomicField(78))
M = H._module
chi = DirichletCharacter(H, M([13,58]))
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pari:[g,chi] = znchar(Mod(35,4563))
\(\chi_{4563}(35,\cdot)\)
\(\chi_{4563}(341,\cdot)\)
\(\chi_{4563}(386,\cdot)\)
\(\chi_{4563}(692,\cdot)\)
\(\chi_{4563}(737,\cdot)\)
\(\chi_{4563}(1043,\cdot)\)
\(\chi_{4563}(1088,\cdot)\)
\(\chi_{4563}(1394,\cdot)\)
\(\chi_{4563}(1439,\cdot)\)
\(\chi_{4563}(1745,\cdot)\)
\(\chi_{4563}(1790,\cdot)\)
\(\chi_{4563}(2096,\cdot)\)
\(\chi_{4563}(2141,\cdot)\)
\(\chi_{4563}(2447,\cdot)\)
\(\chi_{4563}(2492,\cdot)\)
\(\chi_{4563}(2798,\cdot)\)
\(\chi_{4563}(2843,\cdot)\)
\(\chi_{4563}(3149,\cdot)\)
\(\chi_{4563}(3194,\cdot)\)
\(\chi_{4563}(3500,\cdot)\)
\(\chi_{4563}(3545,\cdot)\)
\(\chi_{4563}(3851,\cdot)\)
\(\chi_{4563}(3896,\cdot)\)
\(\chi_{4563}(4553,\cdot)\)
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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 ]
\((677,3889)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{29}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4563 }(35, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{5}{78}\right)\) |
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sage:chi.jacobi_sum(n)
