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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([48]))
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pari:[g,chi] = znchar(Mod(49,197))
| Modulus: | \(197\) | |
| Conductor: | \(197\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(49\) |
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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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\(\chi_{197}(16,\cdot)\)
\(\chi_{197}(23,\cdot)\)
\(\chi_{197}(24,\cdot)\)
\(\chi_{197}(28,\cdot)\)
\(\chi_{197}(29,\cdot)\)
\(\chi_{197}(34,\cdot)\)
\(\chi_{197}(37,\cdot)\)
\(\chi_{197}(40,\cdot)\)
\(\chi_{197}(42,\cdot)\)
\(\chi_{197}(49,\cdot)\)
\(\chi_{197}(51,\cdot)\)
\(\chi_{197}(53,\cdot)\)
\(\chi_{197}(54,\cdot)\)
\(\chi_{197}(59,\cdot)\)
\(\chi_{197}(60,\cdot)\)
\(\chi_{197}(61,\cdot)\)
\(\chi_{197}(63,\cdot)\)
\(\chi_{197}(70,\cdot)\)
\(\chi_{197}(76,\cdot)\)
\(\chi_{197}(81,\cdot)\)
\(\chi_{197}(85,\cdot)\)
\(\chi_{197}(88,\cdot)\)
\(\chi_{197}(90,\cdot)\)
\(\chi_{197}(100,\cdot)\)
\(\chi_{197}(101,\cdot)\)
\(\chi_{197}(105,\cdot)\)
\(\chi_{197}(132,\cdot)\)
\(\chi_{197}(133,\cdot)\)
\(\chi_{197}(135,\cdot)\)
\(\chi_{197}(142,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{24}{49}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 197 }(49, a) \) |
\(1\) | \(1\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{32}{49}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{15}{49}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{10}{49}\right)\) |
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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)
