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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(763, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([72,77]))
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pari:[g,chi] = znchar(Mod(466,763))
| Modulus: | \(763\) | |
| Conductor: | \(763\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(108\) |
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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)
|
\(\chi_{763}(11,\cdot)\)
\(\chi_{763}(18,\cdot)\)
\(\chi_{763}(39,\cdot)\)
\(\chi_{763}(51,\cdot)\)
\(\chi_{763}(58,\cdot)\)
\(\chi_{763}(67,\cdot)\)
\(\chi_{763}(149,\cdot)\)
\(\chi_{763}(151,\cdot)\)
\(\chi_{763}(179,\cdot)\)
\(\chi_{763}(200,\cdot)\)
\(\chi_{763}(205,\cdot)\)
\(\chi_{763}(207,\cdot)\)
\(\chi_{763}(212,\cdot)\)
\(\chi_{763}(275,\cdot)\)
\(\chi_{763}(277,\cdot)\)
\(\chi_{763}(303,\cdot)\)
\(\chi_{763}(317,\cdot)\)
\(\chi_{763}(380,\cdot)\)
\(\chi_{763}(389,\cdot)\)
\(\chi_{763}(422,\cdot)\)
\(\chi_{763}(450,\cdot)\)
\(\chi_{763}(466,\cdot)\)
\(\chi_{763}(473,\cdot)\)
\(\chi_{763}(480,\cdot)\)
\(\chi_{763}(492,\cdot)\)
\(\chi_{763}(501,\cdot)\)
\(\chi_{763}(508,\cdot)\)
\(\chi_{763}(515,\cdot)\)
\(\chi_{763}(555,\cdot)\)
\(\chi_{763}(569,\cdot)\)
...
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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 ]
\((437,442)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{77}{108}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 763 }(466, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{37}{54}\right)\) |
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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)
