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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(107, base_ring=CyclotomicField(106))
M = H._module
chi = DirichletCharacter(H, M([32]))
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pari:[g,chi] = znchar(Mod(29,107))
| Modulus: | \(107\) | |
| Conductor: | \(107\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(53\) |
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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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\(\chi_{107}(3,\cdot)\)
\(\chi_{107}(4,\cdot)\)
\(\chi_{107}(9,\cdot)\)
\(\chi_{107}(10,\cdot)\)
\(\chi_{107}(11,\cdot)\)
\(\chi_{107}(12,\cdot)\)
\(\chi_{107}(13,\cdot)\)
\(\chi_{107}(14,\cdot)\)
\(\chi_{107}(16,\cdot)\)
\(\chi_{107}(19,\cdot)\)
\(\chi_{107}(23,\cdot)\)
\(\chi_{107}(25,\cdot)\)
\(\chi_{107}(27,\cdot)\)
\(\chi_{107}(29,\cdot)\)
\(\chi_{107}(30,\cdot)\)
\(\chi_{107}(33,\cdot)\)
\(\chi_{107}(34,\cdot)\)
\(\chi_{107}(35,\cdot)\)
\(\chi_{107}(36,\cdot)\)
\(\chi_{107}(37,\cdot)\)
\(\chi_{107}(39,\cdot)\)
\(\chi_{107}(40,\cdot)\)
\(\chi_{107}(41,\cdot)\)
\(\chi_{107}(42,\cdot)\)
\(\chi_{107}(44,\cdot)\)
\(\chi_{107}(47,\cdot)\)
\(\chi_{107}(48,\cdot)\)
\(\chi_{107}(49,\cdot)\)
\(\chi_{107}(52,\cdot)\)
\(\chi_{107}(53,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{16}{53}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 107 }(29, a) \) |
\(1\) | \(1\) | \(e\left(\frac{16}{53}\right)\) | \(e\left(\frac{7}{53}\right)\) | \(e\left(\frac{32}{53}\right)\) | \(e\left(\frac{10}{53}\right)\) | \(e\left(\frac{23}{53}\right)\) | \(e\left(\frac{52}{53}\right)\) | \(e\left(\frac{48}{53}\right)\) | \(e\left(\frac{14}{53}\right)\) | \(e\left(\frac{26}{53}\right)\) | \(e\left(\frac{34}{53}\right)\) |
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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)
