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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2450, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([119,130]))
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pari:[g,chi] = znchar(Mod(1147,2450))
\(\chi_{2450}(13,\cdot)\)
\(\chi_{2450}(27,\cdot)\)
\(\chi_{2450}(83,\cdot)\)
\(\chi_{2450}(153,\cdot)\)
\(\chi_{2450}(167,\cdot)\)
\(\chi_{2450}(223,\cdot)\)
\(\chi_{2450}(237,\cdot)\)
\(\chi_{2450}(363,\cdot)\)
\(\chi_{2450}(377,\cdot)\)
\(\chi_{2450}(433,\cdot)\)
\(\chi_{2450}(447,\cdot)\)
\(\chi_{2450}(503,\cdot)\)
\(\chi_{2450}(517,\cdot)\)
\(\chi_{2450}(573,\cdot)\)
\(\chi_{2450}(713,\cdot)\)
\(\chi_{2450}(727,\cdot)\)
\(\chi_{2450}(797,\cdot)\)
\(\chi_{2450}(853,\cdot)\)
\(\chi_{2450}(867,\cdot)\)
\(\chi_{2450}(923,\cdot)\)
\(\chi_{2450}(937,\cdot)\)
\(\chi_{2450}(1063,\cdot)\)
\(\chi_{2450}(1133,\cdot)\)
\(\chi_{2450}(1147,\cdot)\)
\(\chi_{2450}(1203,\cdot)\)
\(\chi_{2450}(1217,\cdot)\)
\(\chi_{2450}(1287,\cdot)\)
\(\chi_{2450}(1413,\cdot)\)
\(\chi_{2450}(1427,\cdot)\)
\(\chi_{2450}(1483,\cdot)\)
...
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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 ]
\((1177,101)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{13}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 2450 }(1147, a) \) |
\(1\) | \(1\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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sage:chi.jacobi_sum(n)
