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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(205, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([30,17]))
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pari:[g,chi] = znchar(Mod(108,205))
| Modulus: | \(205\) | |
| Conductor: | \(205\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(40\) |
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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
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| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{205}(7,\cdot)\)
\(\chi_{205}(12,\cdot)\)
\(\chi_{205}(52,\cdot)\)
\(\chi_{205}(58,\cdot)\)
\(\chi_{205}(63,\cdot)\)
\(\chi_{205}(88,\cdot)\)
\(\chi_{205}(108,\cdot)\)
\(\chi_{205}(112,\cdot)\)
\(\chi_{205}(138,\cdot)\)
\(\chi_{205}(152,\cdot)\)
\(\chi_{205}(157,\cdot)\)
\(\chi_{205}(158,\cdot)\)
\(\chi_{205}(177,\cdot)\)
\(\chi_{205}(183,\cdot)\)
\(\chi_{205}(188,\cdot)\)
\(\chi_{205}(192,\cdot)\)
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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 ]
\((42,6)\) → \((-i,e\left(\frac{17}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 205 }(108, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) |
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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)
