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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(164025, base_ring=CyclotomicField(8748))
M = H._module
chi = DirichletCharacter(H, M([8336,2187]))
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pari:[g,chi] = znchar(Mod(457,164025))
\(\chi_{164025}(7,\cdot)\)
\(\chi_{164025}(43,\cdot)\)
\(\chi_{164025}(157,\cdot)\)
\(\chi_{164025}(193,\cdot)\)
\(\chi_{164025}(232,\cdot)\)
\(\chi_{164025}(268,\cdot)\)
\(\chi_{164025}(382,\cdot)\)
\(\chi_{164025}(418,\cdot)\)
\(\chi_{164025}(457,\cdot)\)
\(\chi_{164025}(493,\cdot)\)
\(\chi_{164025}(607,\cdot)\)
\(\chi_{164025}(643,\cdot)\)
\(\chi_{164025}(682,\cdot)\)
\(\chi_{164025}(718,\cdot)\)
\(\chi_{164025}(832,\cdot)\)
\(\chi_{164025}(868,\cdot)\)
\(\chi_{164025}(907,\cdot)\)
\(\chi_{164025}(943,\cdot)\)
\(\chi_{164025}(1057,\cdot)\)
\(\chi_{164025}(1093,\cdot)\)
\(\chi_{164025}(1132,\cdot)\)
\(\chi_{164025}(1168,\cdot)\)
\(\chi_{164025}(1282,\cdot)\)
\(\chi_{164025}(1318,\cdot)\)
\(\chi_{164025}(1357,\cdot)\)
\(\chi_{164025}(1393,\cdot)\)
\(\chi_{164025}(1507,\cdot)\)
\(\chi_{164025}(1543,\cdot)\)
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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 ]
\((59051,104977)\) → \((e\left(\frac{2084}{2187}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 164025 }(457, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1775}{8748}\right)\) | \(e\left(\frac{1775}{4374}\right)\) | \(e\left(\frac{239}{8748}\right)\) | \(e\left(\frac{1775}{2916}\right)\) | \(e\left(\frac{983}{2187}\right)\) | \(e\left(\frac{5857}{8748}\right)\) | \(e\left(\frac{1007}{4374}\right)\) | \(e\left(\frac{1775}{2187}\right)\) | \(e\left(\frac{1057}{2916}\right)\) | \(e\left(\frac{763}{1458}\right)\) |
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sage:chi.jacobi_sum(n)
