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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([0,210,147,370]))
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pari:[g,chi] = znchar(Mod(10853,44100))
\(\chi_{44100}(17,\cdot)\)
\(\chi_{44100}(773,\cdot)\)
\(\chi_{44100}(1277,\cdot)\)
\(\chi_{44100}(1853,\cdot)\)
\(\chi_{44100}(2033,\cdot)\)
\(\chi_{44100}(2537,\cdot)\)
\(\chi_{44100}(3113,\cdot)\)
\(\chi_{44100}(3617,\cdot)\)
\(\chi_{44100}(3797,\cdot)\)
\(\chi_{44100}(4373,\cdot)\)
\(\chi_{44100}(4553,\cdot)\)
\(\chi_{44100}(4877,\cdot)\)
\(\chi_{44100}(5633,\cdot)\)
\(\chi_{44100}(6137,\cdot)\)
\(\chi_{44100}(6317,\cdot)\)
\(\chi_{44100}(7073,\cdot)\)
\(\chi_{44100}(7397,\cdot)\)
\(\chi_{44100}(8333,\cdot)\)
\(\chi_{44100}(8837,\cdot)\)
\(\chi_{44100}(9413,\cdot)\)
\(\chi_{44100}(10097,\cdot)\)
\(\chi_{44100}(10673,\cdot)\)
\(\chi_{44100}(10853,\cdot)\)
\(\chi_{44100}(11177,\cdot)\)
\(\chi_{44100}(11933,\cdot)\)
\(\chi_{44100}(12113,\cdot)\)
\(\chi_{44100}(12437,\cdot)\)
\(\chi_{44100}(12617,\cdot)\)
\(\chi_{44100}(13373,\cdot)\)
\(\chi_{44100}(13697,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((1,-1,e\left(\frac{7}{20}\right),e\left(\frac{37}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(10853, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{347}{420}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{143}{420}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{15}{28}\right)\) |
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sage:chi.jacobi_sum(n)
