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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,5]))
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pari:[g,chi] = znchar(Mod(32,555))
| Modulus: | \(555\) | |
| Conductor: | \(555\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(36\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{555}(2,\cdot)\)
\(\chi_{555}(32,\cdot)\)
\(\chi_{555}(92,\cdot)\)
\(\chi_{555}(128,\cdot)\)
\(\chi_{555}(242,\cdot)\)
\(\chi_{555}(278,\cdot)\)
\(\chi_{555}(338,\cdot)\)
\(\chi_{555}(368,\cdot)\)
\(\chi_{555}(383,\cdot)\)
\(\chi_{555}(422,\cdot)\)
\(\chi_{555}(503,\cdot)\)
\(\chi_{555}(542,\cdot)\)
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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 ]
\((371,112,76)\) → \((-1,i,e\left(\frac{5}{36}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 555 }(32, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
