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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14700, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,0,84,95]))
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pari:[g,chi] = znchar(Mod(5281,14700))
\(\chi_{14700}(61,\cdot)\)
\(\chi_{14700}(241,\cdot)\)
\(\chi_{14700}(481,\cdot)\)
\(\chi_{14700}(661,\cdot)\)
\(\chi_{14700}(1081,\cdot)\)
\(\chi_{14700}(1321,\cdot)\)
\(\chi_{14700}(1741,\cdot)\)
\(\chi_{14700}(1921,\cdot)\)
\(\chi_{14700}(2161,\cdot)\)
\(\chi_{14700}(2341,\cdot)\)
\(\chi_{14700}(2581,\cdot)\)
\(\chi_{14700}(2761,\cdot)\)
\(\chi_{14700}(3181,\cdot)\)
\(\chi_{14700}(3421,\cdot)\)
\(\chi_{14700}(4021,\cdot)\)
\(\chi_{14700}(4261,\cdot)\)
\(\chi_{14700}(4681,\cdot)\)
\(\chi_{14700}(4861,\cdot)\)
\(\chi_{14700}(5281,\cdot)\)
\(\chi_{14700}(5521,\cdot)\)
\(\chi_{14700}(5941,\cdot)\)
\(\chi_{14700}(6121,\cdot)\)
\(\chi_{14700}(6361,\cdot)\)
\(\chi_{14700}(6541,\cdot)\)
\(\chi_{14700}(6961,\cdot)\)
\(\chi_{14700}(7621,\cdot)\)
\(\chi_{14700}(8041,\cdot)\)
\(\chi_{14700}(8221,\cdot)\)
\(\chi_{14700}(8461,\cdot)\)
\(\chi_{14700}(8641,\cdot)\)
...
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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 ]
\((7351,4901,1177,9901)\) → \((1,1,e\left(\frac{2}{5}\right),e\left(\frac{19}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14700 }(5281, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{5}{7}\right)\) |
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sage:chi.jacobi_sum(n)
