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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6223, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([3,1]))
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pari:[g,chi] = znchar(Mod(27,6223))
| Modulus: | \(6223\) | |
| Conductor: | \(6223\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(42\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{6223}(27,\cdot)\)
\(\chi_{6223}(461,\cdot)\)
\(\chi_{6223}(559,\cdot)\)
\(\chi_{6223}(1945,\cdot)\)
\(\chi_{6223}(1994,\cdot)\)
\(\chi_{6223}(2169,\cdot)\)
\(\chi_{6223}(3737,\cdot)\)
\(\chi_{6223}(4892,\cdot)\)
\(\chi_{6223}(5284,\cdot)\)
\(\chi_{6223}(5466,\cdot)\)
\(\chi_{6223}(5494,\cdot)\)
\(\chi_{6223}(5690,\cdot)\)
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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 ]
\((5589,638)\) → \((e\left(\frac{1}{14}\right),e\left(\frac{1}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 6223 }(27, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) |
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sage:chi.jacobi_sum(n)
