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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(459, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([56,45]))
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pari:[g,chi] = znchar(Mod(209,459))
| Modulus: | \(459\) | |
| Conductor: | \(459\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(144\) |
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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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\(\chi_{459}(5,\cdot)\)
\(\chi_{459}(11,\cdot)\)
\(\chi_{459}(14,\cdot)\)
\(\chi_{459}(20,\cdot)\)
\(\chi_{459}(23,\cdot)\)
\(\chi_{459}(29,\cdot)\)
\(\chi_{459}(41,\cdot)\)
\(\chi_{459}(56,\cdot)\)
\(\chi_{459}(65,\cdot)\)
\(\chi_{459}(74,\cdot)\)
\(\chi_{459}(92,\cdot)\)
\(\chi_{459}(95,\cdot)\)
\(\chi_{459}(113,\cdot)\)
\(\chi_{459}(122,\cdot)\)
\(\chi_{459}(131,\cdot)\)
\(\chi_{459}(146,\cdot)\)
\(\chi_{459}(158,\cdot)\)
\(\chi_{459}(164,\cdot)\)
\(\chi_{459}(167,\cdot)\)
\(\chi_{459}(173,\cdot)\)
\(\chi_{459}(176,\cdot)\)
\(\chi_{459}(182,\cdot)\)
\(\chi_{459}(194,\cdot)\)
\(\chi_{459}(209,\cdot)\)
\(\chi_{459}(218,\cdot)\)
\(\chi_{459}(227,\cdot)\)
\(\chi_{459}(245,\cdot)\)
\(\chi_{459}(248,\cdot)\)
\(\chi_{459}(266,\cdot)\)
\(\chi_{459}(275,\cdot)\)
...
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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 ]
\((137,190)\) → \((e\left(\frac{7}{18}\right),e\left(\frac{5}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 459 }(209, a) \) |
\(1\) | \(1\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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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)
