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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5243, base_ring=CyclotomicField(742))
M = H._module
chi = DirichletCharacter(H, M([689,7]))
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pari:[g,chi] = znchar(Mod(216,5243))
| Modulus: | \(5243\) | |
| Conductor: | \(5243\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(742\) |
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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)
|
\(\chi_{5243}(6,\cdot)\)
\(\chi_{5243}(20,\cdot)\)
\(\chi_{5243}(55,\cdot)\)
\(\chi_{5243}(104,\cdot)\)
\(\chi_{5243}(125,\cdot)\)
\(\chi_{5243}(139,\cdot)\)
\(\chi_{5243}(153,\cdot)\)
\(\chi_{5243}(167,\cdot)\)
\(\chi_{5243}(174,\cdot)\)
\(\chi_{5243}(181,\cdot)\)
\(\chi_{5243}(202,\cdot)\)
\(\chi_{5243}(216,\cdot)\)
\(\chi_{5243}(265,\cdot)\)
\(\chi_{5243}(272,\cdot)\)
\(\chi_{5243}(279,\cdot)\)
\(\chi_{5243}(286,\cdot)\)
\(\chi_{5243}(307,\cdot)\)
\(\chi_{5243}(328,\cdot)\)
\(\chi_{5243}(349,\cdot)\)
\(\chi_{5243}(384,\cdot)\)
\(\chi_{5243}(398,\cdot)\)
\(\chi_{5243}(405,\cdot)\)
\(\chi_{5243}(412,\cdot)\)
\(\chi_{5243}(419,\cdot)\)
\(\chi_{5243}(433,\cdot)\)
\(\chi_{5243}(454,\cdot)\)
\(\chi_{5243}(482,\cdot)\)
\(\chi_{5243}(496,\cdot)\)
\(\chi_{5243}(510,\cdot)\)
\(\chi_{5243}(524,\cdot)\)
...
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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 ]
\((2355,4068)\) → \((e\left(\frac{13}{14}\right),e\left(\frac{1}{106}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 5243 }(216, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{742}\right)\) | \(e\left(\frac{437}{742}\right)\) | \(e\left(\frac{113}{371}\right)\) | \(e\left(\frac{138}{371}\right)\) | \(e\left(\frac{275}{371}\right)\) | \(e\left(\frac{339}{742}\right)\) | \(e\left(\frac{66}{371}\right)\) | \(e\left(\frac{389}{742}\right)\) | \(e\left(\frac{130}{371}\right)\) | \(e\left(\frac{663}{742}\right)\) |
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sage:chi.jacobi_sum(n)
