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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4477, base_ring=CyclotomicField(660))
M = H._module
chi = DirichletCharacter(H, M([174,385]))
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pari:[g,chi] = znchar(Mod(325,4477))
| Modulus: | \(4477\) | |
| Conductor: | \(4477\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(660\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{4477}(8,\cdot)\)
\(\chi_{4477}(29,\cdot)\)
\(\chi_{4477}(51,\cdot)\)
\(\chi_{4477}(134,\cdot)\)
\(\chi_{4477}(140,\cdot)\)
\(\chi_{4477}(156,\cdot)\)
\(\chi_{4477}(162,\cdot)\)
\(\chi_{4477}(171,\cdot)\)
\(\chi_{4477}(193,\cdot)\)
\(\chi_{4477}(288,\cdot)\)
\(\chi_{4477}(304,\cdot)\)
\(\chi_{4477}(310,\cdot)\)
\(\chi_{4477}(325,\cdot)\)
\(\chi_{4477}(347,\cdot)\)
\(\chi_{4477}(393,\cdot)\)
\(\chi_{4477}(415,\cdot)\)
\(\chi_{4477}(436,\cdot)\)
\(\chi_{4477}(458,\cdot)\)
\(\chi_{4477}(541,\cdot)\)
\(\chi_{4477}(547,\cdot)\)
\(\chi_{4477}(563,\cdot)\)
\(\chi_{4477}(569,\cdot)\)
\(\chi_{4477}(600,\cdot)\)
\(\chi_{4477}(689,\cdot)\)
\(\chi_{4477}(695,\cdot)\)
\(\chi_{4477}(711,\cdot)\)
\(\chi_{4477}(732,\cdot)\)
\(\chi_{4477}(754,\cdot)\)
\(\chi_{4477}(800,\cdot)\)
\(\chi_{4477}(822,\cdot)\)
...
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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 ]
\((1333,3147)\) → \((e\left(\frac{29}{110}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 4477 }(325, a) \) |
\(1\) | \(1\) | \(e\left(\frac{559}{660}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{229}{330}\right)\) | \(e\left(\frac{611}{660}\right)\) | \(e\left(\frac{47}{220}\right)\) | \(e\left(\frac{169}{330}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) |
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sage:chi.jacobi_sum(n)
