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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([24]))
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pari:[g,chi] = znchar(Mod(82,121))
| Modulus: | \(121\) | |
| Conductor: | \(121\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(55\) |
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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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\(\chi_{121}(4,\cdot)\)
\(\chi_{121}(5,\cdot)\)
\(\chi_{121}(14,\cdot)\)
\(\chi_{121}(15,\cdot)\)
\(\chi_{121}(16,\cdot)\)
\(\chi_{121}(20,\cdot)\)
\(\chi_{121}(25,\cdot)\)
\(\chi_{121}(26,\cdot)\)
\(\chi_{121}(31,\cdot)\)
\(\chi_{121}(36,\cdot)\)
\(\chi_{121}(37,\cdot)\)
\(\chi_{121}(38,\cdot)\)
\(\chi_{121}(42,\cdot)\)
\(\chi_{121}(47,\cdot)\)
\(\chi_{121}(48,\cdot)\)
\(\chi_{121}(49,\cdot)\)
\(\chi_{121}(53,\cdot)\)
\(\chi_{121}(58,\cdot)\)
\(\chi_{121}(59,\cdot)\)
\(\chi_{121}(60,\cdot)\)
\(\chi_{121}(64,\cdot)\)
\(\chi_{121}(69,\cdot)\)
\(\chi_{121}(70,\cdot)\)
\(\chi_{121}(71,\cdot)\)
\(\chi_{121}(75,\cdot)\)
\(\chi_{121}(80,\cdot)\)
\(\chi_{121}(82,\cdot)\)
\(\chi_{121}(86,\cdot)\)
\(\chi_{121}(91,\cdot)\)
\(\chi_{121}(92,\cdot)\)
...
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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 ]
\(2\) → \(e\left(\frac{12}{55}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 121 }(82, a) \) |
\(1\) | \(1\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
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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)
