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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(330))
M = H._module
chi = DirichletCharacter(H, M([327,275]))
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pari:[g,chi] = znchar(Mod(1029,3751))
| Modulus: | \(3751\) | |
| Conductor: | \(3751\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(330\) |
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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)
|
\(\chi_{3751}(6,\cdot)\)
\(\chi_{3751}(57,\cdot)\)
\(\chi_{3751}(68,\cdot)\)
\(\chi_{3751}(150,\cdot)\)
\(\chi_{3751}(305,\cdot)\)
\(\chi_{3751}(316,\cdot)\)
\(\chi_{3751}(347,\cdot)\)
\(\chi_{3751}(398,\cdot)\)
\(\chi_{3751}(409,\cdot)\)
\(\chi_{3751}(491,\cdot)\)
\(\chi_{3751}(502,\cdot)\)
\(\chi_{3751}(646,\cdot)\)
\(\chi_{3751}(657,\cdot)\)
\(\chi_{3751}(677,\cdot)\)
\(\chi_{3751}(688,\cdot)\)
\(\chi_{3751}(739,\cdot)\)
\(\chi_{3751}(750,\cdot)\)
\(\chi_{3751}(832,\cdot)\)
\(\chi_{3751}(843,\cdot)\)
\(\chi_{3751}(987,\cdot)\)
\(\chi_{3751}(998,\cdot)\)
\(\chi_{3751}(1018,\cdot)\)
\(\chi_{3751}(1029,\cdot)\)
\(\chi_{3751}(1091,\cdot)\)
\(\chi_{3751}(1173,\cdot)\)
\(\chi_{3751}(1184,\cdot)\)
\(\chi_{3751}(1339,\cdot)\)
\(\chi_{3751}(1359,\cdot)\)
\(\chi_{3751}(1370,\cdot)\)
\(\chi_{3751}(1421,\cdot)\)
...
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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 ]
\((2543,2421)\) → \((e\left(\frac{109}{110}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(1029, a) \) |
\(1\) | \(1\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{164}{165}\right)\) | \(e\left(\frac{4}{165}\right)\) | \(e\left(\frac{89}{330}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) |
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sage:chi.jacobi_sum(n)
