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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2178, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([275,177]))
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pari:[g,chi] = znchar(Mod(347,2178))
\(\chi_{2178}(29,\cdot)\)
\(\chi_{2178}(41,\cdot)\)
\(\chi_{2178}(83,\cdot)\)
\(\chi_{2178}(95,\cdot)\)
\(\chi_{2178}(101,\cdot)\)
\(\chi_{2178}(149,\cdot)\)
\(\chi_{2178}(167,\cdot)\)
\(\chi_{2178}(173,\cdot)\)
\(\chi_{2178}(227,\cdot)\)
\(\chi_{2178}(281,\cdot)\)
\(\chi_{2178}(293,\cdot)\)
\(\chi_{2178}(299,\cdot)\)
\(\chi_{2178}(347,\cdot)\)
\(\chi_{2178}(365,\cdot)\)
\(\chi_{2178}(371,\cdot)\)
\(\chi_{2178}(425,\cdot)\)
\(\chi_{2178}(437,\cdot)\)
\(\chi_{2178}(479,\cdot)\)
\(\chi_{2178}(491,\cdot)\)
\(\chi_{2178}(497,\cdot)\)
\(\chi_{2178}(545,\cdot)\)
\(\chi_{2178}(563,\cdot)\)
\(\chi_{2178}(569,\cdot)\)
\(\chi_{2178}(623,\cdot)\)
\(\chi_{2178}(635,\cdot)\)
\(\chi_{2178}(677,\cdot)\)
\(\chi_{2178}(689,\cdot)\)
\(\chi_{2178}(695,\cdot)\)
\(\chi_{2178}(743,\cdot)\)
\(\chi_{2178}(761,\cdot)\)
...
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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 ]
\((1937,1333)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{59}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2178 }(347, a) \) |
\(1\) | \(1\) | \(e\left(\frac{283}{330}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{277}{330}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{118}{165}\right)\) | \(e\left(\frac{157}{165}\right)\) | \(e\left(\frac{131}{165}\right)\) | \(e\left(\frac{52}{55}\right)\) |
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sage:chi.jacobi_sum(n)
