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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1576, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([49,0,15]))
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pari:[g,chi] = znchar(Mod(1007,1576))
\(\chi_{1576}(7,\cdot)\)
\(\chi_{1576}(15,\cdot)\)
\(\chi_{1576}(39,\cdot)\)
\(\chi_{1576}(47,\cdot)\)
\(\chi_{1576}(55,\cdot)\)
\(\chi_{1576}(127,\cdot)\)
\(\chi_{1576}(143,\cdot)\)
\(\chi_{1576}(207,\cdot)\)
\(\chi_{1576}(223,\cdot)\)
\(\chi_{1576}(335,\cdot)\)
\(\chi_{1576}(343,\cdot)\)
\(\chi_{1576}(503,\cdot)\)
\(\chi_{1576}(551,\cdot)\)
\(\chi_{1576}(567,\cdot)\)
\(\chi_{1576}(575,\cdot)\)
\(\chi_{1576}(655,\cdot)\)
\(\chi_{1576}(687,\cdot)\)
\(\chi_{1576}(703,\cdot)\)
\(\chi_{1576}(727,\cdot)\)
\(\chi_{1576}(735,\cdot)\)
\(\chi_{1576}(751,\cdot)\)
\(\chi_{1576}(759,\cdot)\)
\(\chi_{1576}(831,\cdot)\)
\(\chi_{1576}(895,\cdot)\)
\(\chi_{1576}(943,\cdot)\)
\(\chi_{1576}(951,\cdot)\)
\(\chi_{1576}(1007,\cdot)\)
\(\chi_{1576}(1047,\cdot)\)
\(\chi_{1576}(1119,\cdot)\)
\(\chi_{1576}(1159,\cdot)\)
...
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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 ]
\((1183,789,593)\) → \((-1,1,e\left(\frac{15}{98}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1576 }(1007, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{61}{98}\right)\) | \(e\left(\frac{83}{98}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{81}{98}\right)\) | \(e\left(\frac{33}{98}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{98}\right)\) |
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sage:chi.jacobi_sum(n)
