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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6304, base_ring=CyclotomicField(196))
M = H._module
chi = DirichletCharacter(H, M([0,49,139]))
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pari:[g,chi] = znchar(Mod(57,6304))
\(\chi_{6304}(57,\cdot)\)
\(\chi_{6304}(73,\cdot)\)
\(\chi_{6304}(89,\cdot)\)
\(\chi_{6304}(249,\cdot)\)
\(\chi_{6304}(425,\cdot)\)
\(\chi_{6304}(473,\cdot)\)
\(\chi_{6304}(553,\cdot)\)
\(\chi_{6304}(649,\cdot)\)
\(\chi_{6304}(713,\cdot)\)
\(\chi_{6304}(953,\cdot)\)
\(\chi_{6304}(1033,\cdot)\)
\(\chi_{6304}(1065,\cdot)\)
\(\chi_{6304}(1161,\cdot)\)
\(\chi_{6304}(1177,\cdot)\)
\(\chi_{6304}(1193,\cdot)\)
\(\chi_{6304}(1209,\cdot)\)
\(\chi_{6304}(1273,\cdot)\)
\(\chi_{6304}(1305,\cdot)\)
\(\chi_{6304}(1465,\cdot)\)
\(\chi_{6304}(1481,\cdot)\)
\(\chi_{6304}(1593,\cdot)\)
\(\chi_{6304}(1785,\cdot)\)
\(\chi_{6304}(1817,\cdot)\)
\(\chi_{6304}(2041,\cdot)\)
\(\chi_{6304}(2089,\cdot)\)
\(\chi_{6304}(2233,\cdot)\)
\(\chi_{6304}(2297,\cdot)\)
\(\chi_{6304}(2361,\cdot)\)
\(\chi_{6304}(2377,\cdot)\)
\(\chi_{6304}(2489,\cdot)\)
...
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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 ]
\((1183,3941,3745)\) → \((1,i,e\left(\frac{139}{196}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 6304 }(57, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{98}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{47}{98}\right)\) | \(e\left(\frac{149}{196}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{15}{98}\right)\) |
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sage:chi.jacobi_sum(n)
