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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([11]))
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pari:[g,chi] = znchar(Mod(48,125))
| Modulus: | \(125\) | |
| Conductor: | \(125\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(100\) |
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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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\(\chi_{125}(2,\cdot)\)
\(\chi_{125}(3,\cdot)\)
\(\chi_{125}(8,\cdot)\)
\(\chi_{125}(12,\cdot)\)
\(\chi_{125}(13,\cdot)\)
\(\chi_{125}(17,\cdot)\)
\(\chi_{125}(22,\cdot)\)
\(\chi_{125}(23,\cdot)\)
\(\chi_{125}(27,\cdot)\)
\(\chi_{125}(28,\cdot)\)
\(\chi_{125}(33,\cdot)\)
\(\chi_{125}(37,\cdot)\)
\(\chi_{125}(38,\cdot)\)
\(\chi_{125}(42,\cdot)\)
\(\chi_{125}(47,\cdot)\)
\(\chi_{125}(48,\cdot)\)
\(\chi_{125}(52,\cdot)\)
\(\chi_{125}(53,\cdot)\)
\(\chi_{125}(58,\cdot)\)
\(\chi_{125}(62,\cdot)\)
\(\chi_{125}(63,\cdot)\)
\(\chi_{125}(67,\cdot)\)
\(\chi_{125}(72,\cdot)\)
\(\chi_{125}(73,\cdot)\)
\(\chi_{125}(77,\cdot)\)
\(\chi_{125}(78,\cdot)\)
\(\chi_{125}(83,\cdot)\)
\(\chi_{125}(87,\cdot)\)
\(\chi_{125}(88,\cdot)\)
\(\chi_{125}(92,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{11}{100}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 125 }(48, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{27}{50}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{29}{100}\right)\) |
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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)
