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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([52,9]))
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pari:[g,chi] = znchar(Mod(628,1053))
| Modulus: | \(1053\) | |
| Conductor: | \(1053\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(54\) |
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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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\(\chi_{1053}(43,\cdot)\)
\(\chi_{1053}(49,\cdot)\)
\(\chi_{1053}(160,\cdot)\)
\(\chi_{1053}(166,\cdot)\)
\(\chi_{1053}(277,\cdot)\)
\(\chi_{1053}(283,\cdot)\)
\(\chi_{1053}(394,\cdot)\)
\(\chi_{1053}(400,\cdot)\)
\(\chi_{1053}(511,\cdot)\)
\(\chi_{1053}(517,\cdot)\)
\(\chi_{1053}(628,\cdot)\)
\(\chi_{1053}(634,\cdot)\)
\(\chi_{1053}(745,\cdot)\)
\(\chi_{1053}(751,\cdot)\)
\(\chi_{1053}(862,\cdot)\)
\(\chi_{1053}(868,\cdot)\)
\(\chi_{1053}(979,\cdot)\)
\(\chi_{1053}(985,\cdot)\)
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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 ]
\((326,730)\) → \((e\left(\frac{26}{27}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1053 }(628, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) |
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sage:chi.jacobi_sum(n)
