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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(935, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([20,16,65]))
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pari:[g,chi] = znchar(Mod(862,935))
| Modulus: | \(935\) | |
| Conductor: | \(935\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(80\) |
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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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\(\chi_{935}(37,\cdot)\)
\(\chi_{935}(58,\cdot)\)
\(\chi_{935}(82,\cdot)\)
\(\chi_{935}(97,\cdot)\)
\(\chi_{935}(108,\cdot)\)
\(\chi_{935}(113,\cdot)\)
\(\chi_{935}(163,\cdot)\)
\(\chi_{935}(192,\cdot)\)
\(\chi_{935}(207,\cdot)\)
\(\chi_{935}(267,\cdot)\)
\(\chi_{935}(278,\cdot)\)
\(\chi_{935}(313,\cdot)\)
\(\chi_{935}(333,\cdot)\)
\(\chi_{935}(368,\cdot)\)
\(\chi_{935}(377,\cdot)\)
\(\chi_{935}(422,\cdot)\)
\(\chi_{935}(522,\cdot)\)
\(\chi_{935}(532,\cdot)\)
\(\chi_{935}(533,\cdot)\)
\(\chi_{935}(588,\cdot)\)
\(\chi_{935}(592,\cdot)\)
\(\chi_{935}(632,\cdot)\)
\(\chi_{935}(653,\cdot)\)
\(\chi_{935}(702,\cdot)\)
\(\chi_{935}(708,\cdot)\)
\(\chi_{935}(762,\cdot)\)
\(\chi_{935}(823,\cdot)\)
\(\chi_{935}(862,\cdot)\)
\(\chi_{935}(872,\cdot)\)
\(\chi_{935}(873,\cdot)\)
...
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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 ]
\((562,596,496)\) → \((i,e\left(\frac{1}{5}\right),e\left(\frac{13}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 935 }(862, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{33}{80}\right)\) |
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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)
