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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(101, base_ring=CyclotomicField(100))
M = H._module
chi = DirichletCharacter(H, M([11]))
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pari:[g,chi] = znchar(Mod(28,101))
| Modulus: | \(101\) | |
| Conductor: | \(101\) |
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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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\(\chi_{101}(2,\cdot)\)
\(\chi_{101}(3,\cdot)\)
\(\chi_{101}(7,\cdot)\)
\(\chi_{101}(8,\cdot)\)
\(\chi_{101}(11,\cdot)\)
\(\chi_{101}(12,\cdot)\)
\(\chi_{101}(15,\cdot)\)
\(\chi_{101}(18,\cdot)\)
\(\chi_{101}(26,\cdot)\)
\(\chi_{101}(27,\cdot)\)
\(\chi_{101}(28,\cdot)\)
\(\chi_{101}(29,\cdot)\)
\(\chi_{101}(34,\cdot)\)
\(\chi_{101}(35,\cdot)\)
\(\chi_{101}(38,\cdot)\)
\(\chi_{101}(40,\cdot)\)
\(\chi_{101}(42,\cdot)\)
\(\chi_{101}(46,\cdot)\)
\(\chi_{101}(48,\cdot)\)
\(\chi_{101}(50,\cdot)\)
\(\chi_{101}(51,\cdot)\)
\(\chi_{101}(53,\cdot)\)
\(\chi_{101}(55,\cdot)\)
\(\chi_{101}(59,\cdot)\)
\(\chi_{101}(61,\cdot)\)
\(\chi_{101}(63,\cdot)\)
\(\chi_{101}(66,\cdot)\)
\(\chi_{101}(67,\cdot)\)
\(\chi_{101}(72,\cdot)\)
\(\chi_{101}(73,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{11}{100}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 101 }(28, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{100}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{99}{100}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{9}{50}\right)\) | \(-i\) | \(e\left(\frac{43}{100}\right)\) |
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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)
