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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(935, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,12,5]))
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pari:[g,chi] = znchar(Mod(349,935))
| Modulus: | \(935\) | |
| Conductor: | \(935\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(40\) |
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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
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| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{935}(19,\cdot)\)
\(\chi_{935}(94,\cdot)\)
\(\chi_{935}(134,\cdot)\)
\(\chi_{935}(189,\cdot)\)
\(\chi_{935}(304,\cdot)\)
\(\chi_{935}(314,\cdot)\)
\(\chi_{935}(349,\cdot)\)
\(\chi_{935}(359,\cdot)\)
\(\chi_{935}(519,\cdot)\)
\(\chi_{935}(569,\cdot)\)
\(\chi_{935}(644,\cdot)\)
\(\chi_{935}(689,\cdot)\)
\(\chi_{935}(699,\cdot)\)
\(\chi_{935}(739,\cdot)\)
\(\chi_{935}(899,\cdot)\)
\(\chi_{935}(909,\cdot)\)
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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 ]
\((562,596,496)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{1}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 935 }(349, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) |
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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)
