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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10890, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,0,68]))
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pari:[g,chi] = znchar(Mod(3061,10890))
\(\chi_{10890}(91,\cdot)\)
\(\chi_{10890}(181,\cdot)\)
\(\chi_{10890}(361,\cdot)\)
\(\chi_{10890}(631,\cdot)\)
\(\chi_{10890}(1081,\cdot)\)
\(\chi_{10890}(1171,\cdot)\)
\(\chi_{10890}(1351,\cdot)\)
\(\chi_{10890}(1621,\cdot)\)
\(\chi_{10890}(2071,\cdot)\)
\(\chi_{10890}(2161,\cdot)\)
\(\chi_{10890}(2341,\cdot)\)
\(\chi_{10890}(2611,\cdot)\)
\(\chi_{10890}(3061,\cdot)\)
\(\chi_{10890}(3151,\cdot)\)
\(\chi_{10890}(3331,\cdot)\)
\(\chi_{10890}(3601,\cdot)\)
\(\chi_{10890}(4051,\cdot)\)
\(\chi_{10890}(4321,\cdot)\)
\(\chi_{10890}(4591,\cdot)\)
\(\chi_{10890}(5041,\cdot)\)
\(\chi_{10890}(5131,\cdot)\)
\(\chi_{10890}(5311,\cdot)\)
\(\chi_{10890}(5581,\cdot)\)
\(\chi_{10890}(6031,\cdot)\)
\(\chi_{10890}(6121,\cdot)\)
\(\chi_{10890}(6571,\cdot)\)
\(\chi_{10890}(7111,\cdot)\)
\(\chi_{10890}(7291,\cdot)\)
\(\chi_{10890}(7561,\cdot)\)
\(\chi_{10890}(8011,\cdot)\)
...
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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 ]
\((8471,4357,3511)\) → \((1,1,e\left(\frac{34}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 10890 }(3061, a) \) |
\(1\) | \(1\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) |
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sage:chi.jacobi_sum(n)
