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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243, base_ring=CyclotomicField(162))
M = H._module
chi = DirichletCharacter(H, M([31]))
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pari:[g,chi] = znchar(Mod(65,243))
| Modulus: | \(243\) | |
| Conductor: | \(243\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(162\) |
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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
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{243}(2,\cdot)\)
\(\chi_{243}(5,\cdot)\)
\(\chi_{243}(11,\cdot)\)
\(\chi_{243}(14,\cdot)\)
\(\chi_{243}(20,\cdot)\)
\(\chi_{243}(23,\cdot)\)
\(\chi_{243}(29,\cdot)\)
\(\chi_{243}(32,\cdot)\)
\(\chi_{243}(38,\cdot)\)
\(\chi_{243}(41,\cdot)\)
\(\chi_{243}(47,\cdot)\)
\(\chi_{243}(50,\cdot)\)
\(\chi_{243}(56,\cdot)\)
\(\chi_{243}(59,\cdot)\)
\(\chi_{243}(65,\cdot)\)
\(\chi_{243}(68,\cdot)\)
\(\chi_{243}(74,\cdot)\)
\(\chi_{243}(77,\cdot)\)
\(\chi_{243}(83,\cdot)\)
\(\chi_{243}(86,\cdot)\)
\(\chi_{243}(92,\cdot)\)
\(\chi_{243}(95,\cdot)\)
\(\chi_{243}(101,\cdot)\)
\(\chi_{243}(104,\cdot)\)
\(\chi_{243}(110,\cdot)\)
\(\chi_{243}(113,\cdot)\)
\(\chi_{243}(119,\cdot)\)
\(\chi_{243}(122,\cdot)\)
\(\chi_{243}(128,\cdot)\)
\(\chi_{243}(131,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{31}{162}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 243 }(65, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{31}{162}\right)\) | \(e\left(\frac{31}{81}\right)\) | \(e\left(\frac{65}{162}\right)\) | \(e\left(\frac{32}{81}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{25}{162}\right)\) | \(e\left(\frac{43}{81}\right)\) | \(e\left(\frac{95}{162}\right)\) | \(e\left(\frac{62}{81}\right)\) |
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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)
