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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,14]))
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pari:[g,chi] = znchar(Mod(443,507))
| Modulus: | \(507\) | |
| Conductor: | \(507\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(26\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{507}(14,\cdot)\)
\(\chi_{507}(53,\cdot)\)
\(\chi_{507}(92,\cdot)\)
\(\chi_{507}(131,\cdot)\)
\(\chi_{507}(209,\cdot)\)
\(\chi_{507}(248,\cdot)\)
\(\chi_{507}(287,\cdot)\)
\(\chi_{507}(326,\cdot)\)
\(\chi_{507}(365,\cdot)\)
\(\chi_{507}(404,\cdot)\)
\(\chi_{507}(443,\cdot)\)
\(\chi_{507}(482,\cdot)\)
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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 ]
\((170,340)\) → \((-1,e\left(\frac{7}{13}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 507 }(443, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) |
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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)
