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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8954, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([126,88]))
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pari:[g,chi] = znchar(Mod(749,8954))
\(\chi_{8954}(155,\cdot)\)
\(\chi_{8954}(419,\cdot)\)
\(\chi_{8954}(441,\cdot)\)
\(\chi_{8954}(551,\cdot)\)
\(\chi_{8954}(749,\cdot)\)
\(\chi_{8954}(793,\cdot)\)
\(\chi_{8954}(1233,\cdot)\)
\(\chi_{8954}(1255,\cdot)\)
\(\chi_{8954}(1365,\cdot)\)
\(\chi_{8954}(1563,\cdot)\)
\(\chi_{8954}(1607,\cdot)\)
\(\chi_{8954}(1783,\cdot)\)
\(\chi_{8954}(2047,\cdot)\)
\(\chi_{8954}(2069,\cdot)\)
\(\chi_{8954}(2377,\cdot)\)
\(\chi_{8954}(2597,\cdot)\)
\(\chi_{8954}(2861,\cdot)\)
\(\chi_{8954}(2883,\cdot)\)
\(\chi_{8954}(2993,\cdot)\)
\(\chi_{8954}(3191,\cdot)\)
\(\chi_{8954}(3235,\cdot)\)
\(\chi_{8954}(3411,\cdot)\)
\(\chi_{8954}(3675,\cdot)\)
\(\chi_{8954}(3697,\cdot)\)
\(\chi_{8954}(3807,\cdot)\)
\(\chi_{8954}(4005,\cdot)\)
\(\chi_{8954}(4049,\cdot)\)
\(\chi_{8954}(4225,\cdot)\)
\(\chi_{8954}(4489,\cdot)\)
\(\chi_{8954}(4511,\cdot)\)
...
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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 ]
\((1333,3147)\) → \((e\left(\frac{7}{11}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 8954 }(749, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{31}{99}\right)\) | \(e\left(\frac{67}{99}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{16}{99}\right)\) | \(e\left(\frac{86}{99}\right)\) | \(e\left(\frac{29}{99}\right)\) | \(e\left(\frac{37}{99}\right)\) | \(e\left(\frac{23}{99}\right)\) | \(e\left(\frac{7}{33}\right)\) |
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sage:chi.jacobi_sum(n)
