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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8042, base_ring=CyclotomicField(402))
M = H._module
chi = DirichletCharacter(H, M([169]))
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pari:[g,chi] = znchar(Mod(415,8042))
\(\chi_{8042}(53,\cdot)\)
\(\chi_{8042}(147,\cdot)\)
\(\chi_{8042}(153,\cdot)\)
\(\chi_{8042}(245,\cdot)\)
\(\chi_{8042}(249,\cdot)\)
\(\chi_{8042}(251,\cdot)\)
\(\chi_{8042}(255,\cdot)\)
\(\chi_{8042}(287,\cdot)\)
\(\chi_{8042}(331,\cdot)\)
\(\chi_{8042}(341,\cdot)\)
\(\chi_{8042}(381,\cdot)\)
\(\chi_{8042}(415,\cdot)\)
\(\chi_{8042}(425,\cdot)\)
\(\chi_{8042}(439,\cdot)\)
\(\chi_{8042}(543,\cdot)\)
\(\chi_{8042}(549,\cdot)\)
\(\chi_{8042}(557,\cdot)\)
\(\chi_{8042}(599,\cdot)\)
\(\chi_{8042}(635,\cdot)\)
\(\chi_{8042}(717,\cdot)\)
\(\chi_{8042}(733,\cdot)\)
\(\chi_{8042}(801,\cdot)\)
\(\chi_{8042}(841,\cdot)\)
\(\chi_{8042}(859,\cdot)\)
\(\chi_{8042}(905,\cdot)\)
\(\chi_{8042}(915,\cdot)\)
\(\chi_{8042}(983,\cdot)\)
\(\chi_{8042}(1001,\cdot)\)
\(\chi_{8042}(1195,\cdot)\)
\(\chi_{8042}(1211,\cdot)\)
...
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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 ]
\(4023\) → \(e\left(\frac{169}{402}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8042 }(415, a) \) |
\(1\) | \(1\) | \(e\left(\frac{94}{201}\right)\) | \(e\left(\frac{58}{201}\right)\) | \(-1\) | \(e\left(\frac{188}{201}\right)\) | \(e\left(\frac{385}{402}\right)\) | \(e\left(\frac{2}{67}\right)\) | \(e\left(\frac{152}{201}\right)\) | \(e\left(\frac{22}{67}\right)\) | \(e\left(\frac{209}{402}\right)\) | \(e\left(\frac{389}{402}\right)\) |
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sage:chi.jacobi_sum(n)
