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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([0,0,49,30]))
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pari:[g,chi] = znchar(Mod(1009,44100))
\(\chi_{44100}(1009,\cdot)\)
\(\chi_{44100}(2269,\cdot)\)
\(\chi_{44100}(4789,\cdot)\)
\(\chi_{44100}(7309,\cdot)\)
\(\chi_{44100}(8569,\cdot)\)
\(\chi_{44100}(9829,\cdot)\)
\(\chi_{44100}(11089,\cdot)\)
\(\chi_{44100}(13609,\cdot)\)
\(\chi_{44100}(14869,\cdot)\)
\(\chi_{44100}(16129,\cdot)\)
\(\chi_{44100}(17389,\cdot)\)
\(\chi_{44100}(19909,\cdot)\)
\(\chi_{44100}(22429,\cdot)\)
\(\chi_{44100}(23689,\cdot)\)
\(\chi_{44100}(26209,\cdot)\)
\(\chi_{44100}(27469,\cdot)\)
\(\chi_{44100}(28729,\cdot)\)
\(\chi_{44100}(32509,\cdot)\)
\(\chi_{44100}(33769,\cdot)\)
\(\chi_{44100}(35029,\cdot)\)
\(\chi_{44100}(36289,\cdot)\)
\(\chi_{44100}(40069,\cdot)\)
\(\chi_{44100}(41329,\cdot)\)
\(\chi_{44100}(42589,\cdot)\)
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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 ]
\((22051,34301,15877,9901)\) → \((1,1,e\left(\frac{7}{10}\right),e\left(\frac{3}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(1009, a) \) |
\(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{1}{14}\right)\) |
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sage:chi.jacobi_sum(n)
