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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(197, base_ring=CyclotomicField(98))
M = H._module
chi = DirichletCharacter(H, M([5]))
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pari:[g,chi] = znchar(Mod(39,197))
| Modulus: | \(197\) | |
| Conductor: | \(197\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(98\) |
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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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\(\chi_{197}(4,\cdot)\)
\(\chi_{197}(7,\cdot)\)
\(\chi_{197}(9,\cdot)\)
\(\chi_{197}(10,\cdot)\)
\(\chi_{197}(15,\cdot)\)
\(\chi_{197}(22,\cdot)\)
\(\chi_{197}(25,\cdot)\)
\(\chi_{197}(26,\cdot)\)
\(\chi_{197}(39,\cdot)\)
\(\chi_{197}(41,\cdot)\)
\(\chi_{197}(43,\cdot)\)
\(\chi_{197}(47,\cdot)\)
\(\chi_{197}(55,\cdot)\)
\(\chi_{197}(62,\cdot)\)
\(\chi_{197}(64,\cdot)\)
\(\chi_{197}(65,\cdot)\)
\(\chi_{197}(92,\cdot)\)
\(\chi_{197}(96,\cdot)\)
\(\chi_{197}(97,\cdot)\)
\(\chi_{197}(107,\cdot)\)
\(\chi_{197}(109,\cdot)\)
\(\chi_{197}(112,\cdot)\)
\(\chi_{197}(116,\cdot)\)
\(\chi_{197}(121,\cdot)\)
\(\chi_{197}(127,\cdot)\)
\(\chi_{197}(134,\cdot)\)
\(\chi_{197}(136,\cdot)\)
\(\chi_{197}(137,\cdot)\)
\(\chi_{197}(138,\cdot)\)
\(\chi_{197}(143,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{5}{98}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 197 }(39, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{98}\right)\) | \(e\left(\frac{23}{98}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{53}{98}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{15}{98}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{47}{98}\right)\) |
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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)
