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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1275, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,2,35]))
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pari:[g,chi] = znchar(Mod(2,1275))
| Modulus: | \(1275\) | |
| Conductor: | \(1275\) |
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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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\(\chi_{1275}(2,\cdot)\)
\(\chi_{1275}(8,\cdot)\)
\(\chi_{1275}(128,\cdot)\)
\(\chi_{1275}(263,\cdot)\)
\(\chi_{1275}(287,\cdot)\)
\(\chi_{1275}(383,\cdot)\)
\(\chi_{1275}(512,\cdot)\)
\(\chi_{1275}(542,\cdot)\)
\(\chi_{1275}(638,\cdot)\)
\(\chi_{1275}(767,\cdot)\)
\(\chi_{1275}(773,\cdot)\)
\(\chi_{1275}(797,\cdot)\)
\(\chi_{1275}(1022,\cdot)\)
\(\chi_{1275}(1028,\cdot)\)
\(\chi_{1275}(1052,\cdot)\)
\(\chi_{1275}(1148,\cdot)\)
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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 ]
\((851,52,751)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{7}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 1275 }(2, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) |
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sage:chi.jacobi_sum(n)
