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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8042, base_ring=CyclotomicField(670))
M = H._module
chi = DirichletCharacter(H, M([11]))
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pari:[g,chi] = znchar(Mod(445,8042))
\(\chi_{8042}(17,\cdot)\)
\(\chi_{8042}(73,\cdot)\)
\(\chi_{8042}(103,\cdot)\)
\(\chi_{8042}(127,\cdot)\)
\(\chi_{8042}(151,\cdot)\)
\(\chi_{8042}(179,\cdot)\)
\(\chi_{8042}(181,\cdot)\)
\(\chi_{8042}(183,\cdot)\)
\(\chi_{8042}(193,\cdot)\)
\(\chi_{8042}(233,\cdot)\)
\(\chi_{8042}(239,\cdot)\)
\(\chi_{8042}(267,\cdot)\)
\(\chi_{8042}(305,\cdot)\)
\(\chi_{8042}(313,\cdot)\)
\(\chi_{8042}(351,\cdot)\)
\(\chi_{8042}(393,\cdot)\)
\(\chi_{8042}(437,\cdot)\)
\(\chi_{8042}(445,\cdot)\)
\(\chi_{8042}(449,\cdot)\)
\(\chi_{8042}(459,\cdot)\)
\(\chi_{8042}(467,\cdot)\)
\(\chi_{8042}(477,\cdot)\)
\(\chi_{8042}(479,\cdot)\)
\(\chi_{8042}(483,\cdot)\)
\(\chi_{8042}(523,\cdot)\)
\(\chi_{8042}(531,\cdot)\)
\(\chi_{8042}(585,\cdot)\)
\(\chi_{8042}(607,\cdot)\)
\(\chi_{8042}(613,\cdot)\)
\(\chi_{8042}(655,\cdot)\)
...
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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 ]
\(4023\) → \(e\left(\frac{11}{670}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8042 }(445, a) \) |
\(1\) | \(1\) | \(e\left(\frac{332}{335}\right)\) | \(e\left(\frac{202}{335}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{329}{335}\right)\) | \(e\left(\frac{657}{670}\right)\) | \(e\left(\frac{49}{67}\right)\) | \(e\left(\frac{199}{335}\right)\) | \(e\left(\frac{216}{335}\right)\) | \(e\left(\frac{81}{670}\right)\) | \(e\left(\frac{597}{670}\right)\) |
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sage:chi.jacobi_sum(n)
