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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,210,147,400]))
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pari:[g,chi] = znchar(Mod(12653,44100))
\(\chi_{44100}(53,\cdot)\)
\(\chi_{44100}(233,\cdot)\)
\(\chi_{44100}(737,\cdot)\)
\(\chi_{44100}(1313,\cdot)\)
\(\chi_{44100}(1817,\cdot)\)
\(\chi_{44100}(1997,\cdot)\)
\(\chi_{44100}(2573,\cdot)\)
\(\chi_{44100}(2753,\cdot)\)
\(\chi_{44100}(3077,\cdot)\)
\(\chi_{44100}(3833,\cdot)\)
\(\chi_{44100}(4013,\cdot)\)
\(\chi_{44100}(4337,\cdot)\)
\(\chi_{44100}(4517,\cdot)\)
\(\chi_{44100}(5597,\cdot)\)
\(\chi_{44100}(5777,\cdot)\)
\(\chi_{44100}(6353,\cdot)\)
\(\chi_{44100}(6533,\cdot)\)
\(\chi_{44100}(8117,\cdot)\)
\(\chi_{44100}(8297,\cdot)\)
\(\chi_{44100}(8873,\cdot)\)
\(\chi_{44100}(9053,\cdot)\)
\(\chi_{44100}(10133,\cdot)\)
\(\chi_{44100}(10313,\cdot)\)
\(\chi_{44100}(10637,\cdot)\)
\(\chi_{44100}(10817,\cdot)\)
\(\chi_{44100}(11573,\cdot)\)
\(\chi_{44100}(11897,\cdot)\)
\(\chi_{44100}(12077,\cdot)\)
\(\chi_{44100}(12653,\cdot)\)
\(\chi_{44100}(12833,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((1,-1,e\left(\frac{7}{20}\right),e\left(\frac{20}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(12653, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{27}{28}\right)\) |
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sage:chi.jacobi_sum(n)
