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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,9]))
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pari:[g,chi] = znchar(Mod(284,1805))
| Modulus: | \(1805\) | |
| Conductor: | \(1805\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(38\) |
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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)
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\(\chi_{1805}(94,\cdot)\)
\(\chi_{1805}(189,\cdot)\)
\(\chi_{1805}(284,\cdot)\)
\(\chi_{1805}(379,\cdot)\)
\(\chi_{1805}(474,\cdot)\)
\(\chi_{1805}(569,\cdot)\)
\(\chi_{1805}(664,\cdot)\)
\(\chi_{1805}(759,\cdot)\)
\(\chi_{1805}(854,\cdot)\)
\(\chi_{1805}(949,\cdot)\)
\(\chi_{1805}(1044,\cdot)\)
\(\chi_{1805}(1139,\cdot)\)
\(\chi_{1805}(1234,\cdot)\)
\(\chi_{1805}(1329,\cdot)\)
\(\chi_{1805}(1424,\cdot)\)
\(\chi_{1805}(1519,\cdot)\)
\(\chi_{1805}(1614,\cdot)\)
\(\chi_{1805}(1709,\cdot)\)
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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 ]
\((362,1446)\) → \((-1,e\left(\frac{9}{38}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 1805 }(284, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) |
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sage:chi.jacobi_sum(n)
