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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([35,35,42,40]))
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pari:[g,chi] = znchar(Mod(71,44100))
\(\chi_{44100}(71,\cdot)\)
\(\chi_{44100}(1331,\cdot)\)
\(\chi_{44100}(2591,\cdot)\)
\(\chi_{44100}(5111,\cdot)\)
\(\chi_{44100}(7631,\cdot)\)
\(\chi_{44100}(8891,\cdot)\)
\(\chi_{44100}(11411,\cdot)\)
\(\chi_{44100}(12671,\cdot)\)
\(\chi_{44100}(13931,\cdot)\)
\(\chi_{44100}(17711,\cdot)\)
\(\chi_{44100}(18971,\cdot)\)
\(\chi_{44100}(20231,\cdot)\)
\(\chi_{44100}(21491,\cdot)\)
\(\chi_{44100}(25271,\cdot)\)
\(\chi_{44100}(26531,\cdot)\)
\(\chi_{44100}(27791,\cdot)\)
\(\chi_{44100}(30311,\cdot)\)
\(\chi_{44100}(31571,\cdot)\)
\(\chi_{44100}(34091,\cdot)\)
\(\chi_{44100}(36611,\cdot)\)
\(\chi_{44100}(37871,\cdot)\)
\(\chi_{44100}(39131,\cdot)\)
\(\chi_{44100}(40391,\cdot)\)
\(\chi_{44100}(42911,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((-1,-1,e\left(\frac{3}{5}\right),e\left(\frac{4}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(71, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{13}{14}\right)\) |
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sage:chi.jacobi_sum(n)
