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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(783, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([20,9]))
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pari:[g,chi] = znchar(Mod(592,783))
| Modulus: | \(783\) | |
| Conductor: | \(783\) |
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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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\(\chi_{783}(70,\cdot)\)
\(\chi_{783}(133,\cdot)\)
\(\chi_{783}(157,\cdot)\)
\(\chi_{783}(220,\cdot)\)
\(\chi_{783}(331,\cdot)\)
\(\chi_{783}(394,\cdot)\)
\(\chi_{783}(418,\cdot)\)
\(\chi_{783}(481,\cdot)\)
\(\chi_{783}(592,\cdot)\)
\(\chi_{783}(655,\cdot)\)
\(\chi_{783}(679,\cdot)\)
\(\chi_{783}(742,\cdot)\)
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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 ]
\((407,379)\) → \((e\left(\frac{5}{9}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 783 }(592, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) |
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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)
