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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(127, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([64]))
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pari:[g,chi] = znchar(Mod(124,127))
| Modulus: | \(127\) | |
| Conductor: | \(127\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(63\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{127}(9,\cdot)\)
\(\chi_{127}(11,\cdot)\)
\(\chi_{127}(13,\cdot)\)
\(\chi_{127}(15,\cdot)\)
\(\chi_{127}(17,\cdot)\)
\(\chi_{127}(18,\cdot)\)
\(\chi_{127}(21,\cdot)\)
\(\chi_{127}(26,\cdot)\)
\(\chi_{127}(30,\cdot)\)
\(\chi_{127}(31,\cdot)\)
\(\chi_{127}(34,\cdot)\)
\(\chi_{127}(35,\cdot)\)
\(\chi_{127}(36,\cdot)\)
\(\chi_{127}(41,\cdot)\)
\(\chi_{127}(42,\cdot)\)
\(\chi_{127}(44,\cdot)\)
\(\chi_{127}(49,\cdot)\)
\(\chi_{127}(60,\cdot)\)
\(\chi_{127}(62,\cdot)\)
\(\chi_{127}(69,\cdot)\)
\(\chi_{127}(70,\cdot)\)
\(\chi_{127}(71,\cdot)\)
\(\chi_{127}(72,\cdot)\)
\(\chi_{127}(74,\cdot)\)
\(\chi_{127}(79,\cdot)\)
\(\chi_{127}(81,\cdot)\)
\(\chi_{127}(82,\cdot)\)
\(\chi_{127}(84,\cdot)\)
\(\chi_{127}(88,\cdot)\)
\(\chi_{127}(98,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{32}{63}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 127 }(124, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{34}{63}\right)\) |
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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)
