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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44100, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,70,63,40]))
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pari:[g,chi] = znchar(Mod(14989,44100))
\(\chi_{44100}(529,\cdot)\)
\(\chi_{44100}(1129,\cdot)\)
\(\chi_{44100}(1789,\cdot)\)
\(\chi_{44100}(2389,\cdot)\)
\(\chi_{44100}(4309,\cdot)\)
\(\chi_{44100}(4909,\cdot)\)
\(\chi_{44100}(5569,\cdot)\)
\(\chi_{44100}(6169,\cdot)\)
\(\chi_{44100}(8089,\cdot)\)
\(\chi_{44100}(8689,\cdot)\)
\(\chi_{44100}(10609,\cdot)\)
\(\chi_{44100}(11209,\cdot)\)
\(\chi_{44100}(11869,\cdot)\)
\(\chi_{44100}(12469,\cdot)\)
\(\chi_{44100}(13129,\cdot)\)
\(\chi_{44100}(13729,\cdot)\)
\(\chi_{44100}(14389,\cdot)\)
\(\chi_{44100}(14989,\cdot)\)
\(\chi_{44100}(16909,\cdot)\)
\(\chi_{44100}(17509,\cdot)\)
\(\chi_{44100}(18169,\cdot)\)
\(\chi_{44100}(18769,\cdot)\)
\(\chi_{44100}(19429,\cdot)\)
\(\chi_{44100}(20029,\cdot)\)
\(\chi_{44100}(20689,\cdot)\)
\(\chi_{44100}(21289,\cdot)\)
\(\chi_{44100}(23209,\cdot)\)
\(\chi_{44100}(23809,\cdot)\)
\(\chi_{44100}(25729,\cdot)\)
\(\chi_{44100}(26329,\cdot)\)
...
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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 ]
\((22051,34301,15877,9901)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right),e\left(\frac{4}{21}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 44100 }(14989, a) \) |
\(1\) | \(1\) | \(e\left(\frac{79}{105}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{41}{42}\right)\) |
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sage:chi.jacobi_sum(n)
