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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9450, base_ring=CyclotomicField(90))
M = H._module
chi = DirichletCharacter(H, M([5,9,15]))
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pari:[g,chi] = znchar(Mod(6779,9450))
\(\chi_{9450}(479,\cdot)\)
\(\chi_{9450}(509,\cdot)\)
\(\chi_{9450}(1109,\cdot)\)
\(\chi_{9450}(1139,\cdot)\)
\(\chi_{9450}(1739,\cdot)\)
\(\chi_{9450}(1769,\cdot)\)
\(\chi_{9450}(2369,\cdot)\)
\(\chi_{9450}(3029,\cdot)\)
\(\chi_{9450}(3629,\cdot)\)
\(\chi_{9450}(3659,\cdot)\)
\(\chi_{9450}(4259,\cdot)\)
\(\chi_{9450}(4289,\cdot)\)
\(\chi_{9450}(4889,\cdot)\)
\(\chi_{9450}(4919,\cdot)\)
\(\chi_{9450}(5519,\cdot)\)
\(\chi_{9450}(6179,\cdot)\)
\(\chi_{9450}(6779,\cdot)\)
\(\chi_{9450}(6809,\cdot)\)
\(\chi_{9450}(7409,\cdot)\)
\(\chi_{9450}(7439,\cdot)\)
\(\chi_{9450}(8039,\cdot)\)
\(\chi_{9450}(8069,\cdot)\)
\(\chi_{9450}(8669,\cdot)\)
\(\chi_{9450}(9329,\cdot)\)
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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 ]
\((9101,6427,6751)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{1}{10}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 9450 }(6779, a) \) |
\(1\) | \(1\) | \(e\left(\frac{89}{90}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{45}\right)\) | \(e\left(\frac{23}{90}\right)\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{38}{45}\right)\) | \(e\left(\frac{13}{18}\right)\) |
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sage:chi.jacobi_sum(n)
