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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([52,66]))
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pari:[g,chi] = znchar(Mod(937,4563))
\(\chi_{4563}(118,\cdot)\)
\(\chi_{4563}(235,\cdot)\)
\(\chi_{4563}(469,\cdot)\)
\(\chi_{4563}(586,\cdot)\)
\(\chi_{4563}(820,\cdot)\)
\(\chi_{4563}(937,\cdot)\)
\(\chi_{4563}(1171,\cdot)\)
\(\chi_{4563}(1288,\cdot)\)
\(\chi_{4563}(1639,\cdot)\)
\(\chi_{4563}(1873,\cdot)\)
\(\chi_{4563}(1990,\cdot)\)
\(\chi_{4563}(2224,\cdot)\)
\(\chi_{4563}(2341,\cdot)\)
\(\chi_{4563}(2575,\cdot)\)
\(\chi_{4563}(2692,\cdot)\)
\(\chi_{4563}(2926,\cdot)\)
\(\chi_{4563}(3277,\cdot)\)
\(\chi_{4563}(3394,\cdot)\)
\(\chi_{4563}(3628,\cdot)\)
\(\chi_{4563}(3745,\cdot)\)
\(\chi_{4563}(3979,\cdot)\)
\(\chi_{4563}(4096,\cdot)\)
\(\chi_{4563}(4330,\cdot)\)
\(\chi_{4563}(4447,\cdot)\)
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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 ]
\((677,3889)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{11}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4563 }(937, a) \) |
\(1\) | \(1\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{7}{13}\right)\) |
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sage:chi.jacobi_sum(n)
