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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1734, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([0,63]))
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pari:[g,chi] = znchar(Mod(55,1734))
\(\chi_{1734}(13,\cdot)\)
\(\chi_{1734}(55,\cdot)\)
\(\chi_{1734}(115,\cdot)\)
\(\chi_{1734}(157,\cdot)\)
\(\chi_{1734}(217,\cdot)\)
\(\chi_{1734}(259,\cdot)\)
\(\chi_{1734}(319,\cdot)\)
\(\chi_{1734}(361,\cdot)\)
\(\chi_{1734}(421,\cdot)\)
\(\chi_{1734}(463,\cdot)\)
\(\chi_{1734}(523,\cdot)\)
\(\chi_{1734}(565,\cdot)\)
\(\chi_{1734}(625,\cdot)\)
\(\chi_{1734}(667,\cdot)\)
\(\chi_{1734}(727,\cdot)\)
\(\chi_{1734}(769,\cdot)\)
\(\chi_{1734}(871,\cdot)\)
\(\chi_{1734}(931,\cdot)\)
\(\chi_{1734}(973,\cdot)\)
\(\chi_{1734}(1033,\cdot)\)
\(\chi_{1734}(1075,\cdot)\)
\(\chi_{1734}(1135,\cdot)\)
\(\chi_{1734}(1177,\cdot)\)
\(\chi_{1734}(1237,\cdot)\)
\(\chi_{1734}(1279,\cdot)\)
\(\chi_{1734}(1339,\cdot)\)
\(\chi_{1734}(1381,\cdot)\)
\(\chi_{1734}(1441,\cdot)\)
\(\chi_{1734}(1543,\cdot)\)
\(\chi_{1734}(1585,\cdot)\)
...
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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 ]
\((1157,1159)\) → \((1,e\left(\frac{63}{68}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1734 }(55, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) |
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sage:chi.jacobi_sum(n)
