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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([18,10]))
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pari:[g,chi] = znchar(Mod(8041,9025))
\(\chi_{9025}(606,\cdot)\)
\(\chi_{9025}(821,\cdot)\)
\(\chi_{9025}(956,\cdot)\)
\(\chi_{9025}(1111,\cdot)\)
\(\chi_{9025}(1506,\cdot)\)
\(\chi_{9025}(2411,\cdot)\)
\(\chi_{9025}(2581,\cdot)\)
\(\chi_{9025}(2761,\cdot)\)
\(\chi_{9025}(2916,\cdot)\)
\(\chi_{9025}(3311,\cdot)\)
\(\chi_{9025}(4216,\cdot)\)
\(\chi_{9025}(4386,\cdot)\)
\(\chi_{9025}(4431,\cdot)\)
\(\chi_{9025}(4566,\cdot)\)
\(\chi_{9025}(4721,\cdot)\)
\(\chi_{9025}(5116,\cdot)\)
\(\chi_{9025}(6021,\cdot)\)
\(\chi_{9025}(6191,\cdot)\)
\(\chi_{9025}(6236,\cdot)\)
\(\chi_{9025}(6371,\cdot)\)
\(\chi_{9025}(6921,\cdot)\)
\(\chi_{9025}(7996,\cdot)\)
\(\chi_{9025}(8041,\cdot)\)
\(\chi_{9025}(8331,\cdot)\)
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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 ]
\((5777,3251)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(8041, a) \) |
\(1\) | \(1\) | \(e\left(\frac{14}{45}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{28}{45}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{31}{45}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{16}{45}\right)\) |
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sage:chi.jacobi_sum(n)
