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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([6,44]))
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pari:[g,chi] = znchar(Mod(2113,3751))
| Modulus: | \(3751\) | |
| Conductor: | \(3751\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(33\) |
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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)
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\(\chi_{3751}(56,\cdot)\)
\(\chi_{3751}(67,\cdot)\)
\(\chi_{3751}(397,\cdot)\)
\(\chi_{3751}(408,\cdot)\)
\(\chi_{3751}(738,\cdot)\)
\(\chi_{3751}(749,\cdot)\)
\(\chi_{3751}(1079,\cdot)\)
\(\chi_{3751}(1420,\cdot)\)
\(\chi_{3751}(1431,\cdot)\)
\(\chi_{3751}(1761,\cdot)\)
\(\chi_{3751}(1772,\cdot)\)
\(\chi_{3751}(2102,\cdot)\)
\(\chi_{3751}(2113,\cdot)\)
\(\chi_{3751}(2443,\cdot)\)
\(\chi_{3751}(2454,\cdot)\)
\(\chi_{3751}(2795,\cdot)\)
\(\chi_{3751}(3125,\cdot)\)
\(\chi_{3751}(3136,\cdot)\)
\(\chi_{3751}(3466,\cdot)\)
\(\chi_{3751}(3477,\cdot)\)
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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 ]
\((2543,2421)\) → \((e\left(\frac{1}{11}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(2113, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) |
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sage:chi.jacobi_sum(n)
