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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(449, base_ring=CyclotomicField(448))
M = H._module
chi = DirichletCharacter(H, M([439]))
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pari:[g,chi] = znchar(Mod(123,449))
| Modulus: | \(449\) | |
| Conductor: | \(449\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(448\) |
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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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\(\chi_{449}(3,\cdot)\)
\(\chi_{449}(6,\cdot)\)
\(\chi_{449}(12,\cdot)\)
\(\chi_{449}(13,\cdot)\)
\(\chi_{449}(15,\cdot)\)
\(\chi_{449}(17,\cdot)\)
\(\chi_{449}(19,\cdot)\)
\(\chi_{449}(21,\cdot)\)
\(\chi_{449}(26,\cdot)\)
\(\chi_{449}(27,\cdot)\)
\(\chi_{449}(29,\cdot)\)
\(\chi_{449}(30,\cdot)\)
\(\chi_{449}(31,\cdot)\)
\(\chi_{449}(33,\cdot)\)
\(\chi_{449}(34,\cdot)\)
\(\chi_{449}(38,\cdot)\)
\(\chi_{449}(42,\cdot)\)
\(\chi_{449}(43,\cdot)\)
\(\chi_{449}(47,\cdot)\)
\(\chi_{449}(48,\cdot)\)
\(\chi_{449}(54,\cdot)\)
\(\chi_{449}(60,\cdot)\)
\(\chi_{449}(62,\cdot)\)
\(\chi_{449}(65,\cdot)\)
\(\chi_{449}(66,\cdot)\)
\(\chi_{449}(68,\cdot)\)
\(\chi_{449}(69,\cdot)\)
\(\chi_{449}(73,\cdot)\)
\(\chi_{449}(74,\cdot)\)
\(\chi_{449}(75,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{439}{448}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 449 }(123, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{201}{224}\right)\) | \(e\left(\frac{439}{448}\right)\) | \(e\left(\frac{89}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{393}{448}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{155}{224}\right)\) | \(e\left(\frac{215}{224}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{39}{56}\right)\) |
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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)
