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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1740, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,14,14,1]))
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pari:[g,chi] = znchar(Mod(959,1740))
| Modulus: | \(1740\) | |
| Conductor: | \(1740\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(28\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
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sage:chi.is_primitive()
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pari:#znconreyconductor(g,chi)==1
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1740}(119,\cdot)\)
\(\chi_{1740}(359,\cdot)\)
\(\chi_{1740}(479,\cdot)\)
\(\chi_{1740}(599,\cdot)\)
\(\chi_{1740}(659,\cdot)\)
\(\chi_{1740}(839,\cdot)\)
\(\chi_{1740}(959,\cdot)\)
\(\chi_{1740}(1139,\cdot)\)
\(\chi_{1740}(1199,\cdot)\)
\(\chi_{1740}(1319,\cdot)\)
\(\chi_{1740}(1439,\cdot)\)
\(\chi_{1740}(1679,\cdot)\)
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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 ]
\((871,581,697,901)\) → \((-1,-1,-1,e\left(\frac{1}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1740 }(959, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-i\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-i\) | \(e\left(\frac{13}{28}\right)\) |
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sage:chi.jacobi_sum(n)
