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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8464, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,23,33]))
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pari:[g,chi] = znchar(Mod(7911,8464))
\(\chi_{8464}(183,\cdot)\)
\(\chi_{8464}(551,\cdot)\)
\(\chi_{8464}(919,\cdot)\)
\(\chi_{8464}(1287,\cdot)\)
\(\chi_{8464}(1655,\cdot)\)
\(\chi_{8464}(2023,\cdot)\)
\(\chi_{8464}(2391,\cdot)\)
\(\chi_{8464}(2759,\cdot)\)
\(\chi_{8464}(3127,\cdot)\)
\(\chi_{8464}(3495,\cdot)\)
\(\chi_{8464}(3863,\cdot)\)
\(\chi_{8464}(4599,\cdot)\)
\(\chi_{8464}(4967,\cdot)\)
\(\chi_{8464}(5335,\cdot)\)
\(\chi_{8464}(5703,\cdot)\)
\(\chi_{8464}(6071,\cdot)\)
\(\chi_{8464}(6439,\cdot)\)
\(\chi_{8464}(6807,\cdot)\)
\(\chi_{8464}(7175,\cdot)\)
\(\chi_{8464}(7543,\cdot)\)
\(\chi_{8464}(7911,\cdot)\)
\(\chi_{8464}(8279,\cdot)\)
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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 ]
\((7407,2117,6353)\) → \((-1,-1,e\left(\frac{33}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8464 }(7911, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) |
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sage:chi.jacobi_sum(n)
