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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([40,56,55]))
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pari:[g,chi] = znchar(Mod(194,935))
| 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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\(\chi_{935}(24,\cdot)\)
\(\chi_{935}(29,\cdot)\)
\(\chi_{935}(39,\cdot)\)
\(\chi_{935}(74,\cdot)\)
\(\chi_{935}(79,\cdot)\)
\(\chi_{935}(129,\cdot)\)
\(\chi_{935}(139,\cdot)\)
\(\chi_{935}(184,\cdot)\)
\(\chi_{935}(194,\cdot)\)
\(\chi_{935}(244,\cdot)\)
\(\chi_{935}(249,\cdot)\)
\(\chi_{935}(294,\cdot)\)
\(\chi_{935}(299,\cdot)\)
\(\chi_{935}(354,\cdot)\)
\(\chi_{935}(369,\cdot)\)
\(\chi_{935}(414,\cdot)\)
\(\chi_{935}(464,\cdot)\)
\(\chi_{935}(469,\cdot)\)
\(\chi_{935}(479,\cdot)\)
\(\chi_{935}(524,\cdot)\)
\(\chi_{935}(534,\cdot)\)
\(\chi_{935}(589,\cdot)\)
\(\chi_{935}(624,\cdot)\)
\(\chi_{935}(634,\cdot)\)
\(\chi_{935}(734,\cdot)\)
\(\chi_{935}(754,\cdot)\)
\(\chi_{935}(789,\cdot)\)
\(\chi_{935}(794,\cdot)\)
\(\chi_{935}(809,\cdot)\)
\(\chi_{935}(844,\cdot)\)
...
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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 ]
\((562,596,496)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{11}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 935 }(194, a) \) |
\(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) |
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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)
