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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14700, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([0,0,189,320]))
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pari:[g,chi] = znchar(Mod(37,14700))
\(\chi_{14700}(37,\cdot)\)
\(\chi_{14700}(277,\cdot)\)
\(\chi_{14700}(613,\cdot)\)
\(\chi_{14700}(697,\cdot)\)
\(\chi_{14700}(877,\cdot)\)
\(\chi_{14700}(1033,\cdot)\)
\(\chi_{14700}(1117,\cdot)\)
\(\chi_{14700}(1213,\cdot)\)
\(\chi_{14700}(1297,\cdot)\)
\(\chi_{14700}(1453,\cdot)\)
\(\chi_{14700}(1633,\cdot)\)
\(\chi_{14700}(1717,\cdot)\)
\(\chi_{14700}(1873,\cdot)\)
\(\chi_{14700}(2053,\cdot)\)
\(\chi_{14700}(2377,\cdot)\)
\(\chi_{14700}(2473,\cdot)\)
\(\chi_{14700}(2797,\cdot)\)
\(\chi_{14700}(2977,\cdot)\)
\(\chi_{14700}(3133,\cdot)\)
\(\chi_{14700}(3217,\cdot)\)
\(\chi_{14700}(3397,\cdot)\)
\(\chi_{14700}(3553,\cdot)\)
\(\chi_{14700}(3637,\cdot)\)
\(\chi_{14700}(3733,\cdot)\)
\(\chi_{14700}(3817,\cdot)\)
\(\chi_{14700}(3973,\cdot)\)
\(\chi_{14700}(4153,\cdot)\)
\(\chi_{14700}(4237,\cdot)\)
\(\chi_{14700}(4573,\cdot)\)
\(\chi_{14700}(4813,\cdot)\)
...
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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 ]
\((7351,4901,1177,9901)\) → \((1,1,e\left(\frac{9}{20}\right),e\left(\frac{16}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 14700 }(37, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{181}{420}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{9}{28}\right)\) |
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sage:chi.jacobi_sum(n)
