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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2890, base_ring=CyclotomicField(136))
M = H._module
chi = DirichletCharacter(H, M([0,123]))
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pari:[g,chi] = znchar(Mod(831,2890))
\(\chi_{2890}(111,\cdot)\)
\(\chi_{2890}(121,\cdot)\)
\(\chi_{2890}(151,\cdot)\)
\(\chi_{2890}(161,\cdot)\)
\(\chi_{2890}(281,\cdot)\)
\(\chi_{2890}(291,\cdot)\)
\(\chi_{2890}(321,\cdot)\)
\(\chi_{2890}(331,\cdot)\)
\(\chi_{2890}(451,\cdot)\)
\(\chi_{2890}(461,\cdot)\)
\(\chi_{2890}(491,\cdot)\)
\(\chi_{2890}(501,\cdot)\)
\(\chi_{2890}(621,\cdot)\)
\(\chi_{2890}(631,\cdot)\)
\(\chi_{2890}(661,\cdot)\)
\(\chi_{2890}(671,\cdot)\)
\(\chi_{2890}(791,\cdot)\)
\(\chi_{2890}(801,\cdot)\)
\(\chi_{2890}(831,\cdot)\)
\(\chi_{2890}(841,\cdot)\)
\(\chi_{2890}(961,\cdot)\)
\(\chi_{2890}(971,\cdot)\)
\(\chi_{2890}(1011,\cdot)\)
\(\chi_{2890}(1131,\cdot)\)
\(\chi_{2890}(1141,\cdot)\)
\(\chi_{2890}(1171,\cdot)\)
\(\chi_{2890}(1181,\cdot)\)
\(\chi_{2890}(1301,\cdot)\)
\(\chi_{2890}(1341,\cdot)\)
\(\chi_{2890}(1351,\cdot)\)
...
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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 ]
\((1157,581)\) → \((1,e\left(\frac{123}{136}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2890 }(831, a) \) |
\(1\) | \(1\) | \(e\left(\frac{123}{136}\right)\) | \(e\left(\frac{25}{136}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{97}{136}\right)\) | \(e\left(\frac{7}{136}\right)\) |
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sage:chi.jacobi_sum(n)
