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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1127, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([11,133]))
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pari:[g,chi] = znchar(Mod(76,1127))
| Modulus: | \(1127\) | |
| Conductor: | \(1127\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(154\) |
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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)
|
\(\chi_{1127}(20,\cdot)\)
\(\chi_{1127}(34,\cdot)\)
\(\chi_{1127}(76,\cdot)\)
\(\chi_{1127}(83,\cdot)\)
\(\chi_{1127}(90,\cdot)\)
\(\chi_{1127}(111,\cdot)\)
\(\chi_{1127}(125,\cdot)\)
\(\chi_{1127}(132,\cdot)\)
\(\chi_{1127}(153,\cdot)\)
\(\chi_{1127}(181,\cdot)\)
\(\chi_{1127}(237,\cdot)\)
\(\chi_{1127}(251,\cdot)\)
\(\chi_{1127}(258,\cdot)\)
\(\chi_{1127}(272,\cdot)\)
\(\chi_{1127}(286,\cdot)\)
\(\chi_{1127}(314,\cdot)\)
\(\chi_{1127}(356,\cdot)\)
\(\chi_{1127}(398,\cdot)\)
\(\chi_{1127}(405,\cdot)\)
\(\chi_{1127}(412,\cdot)\)
\(\chi_{1127}(419,\cdot)\)
\(\chi_{1127}(433,\cdot)\)
\(\chi_{1127}(447,\cdot)\)
\(\chi_{1127}(454,\cdot)\)
\(\chi_{1127}(475,\cdot)\)
\(\chi_{1127}(503,\cdot)\)
\(\chi_{1127}(517,\cdot)\)
\(\chi_{1127}(559,\cdot)\)
\(\chi_{1127}(566,\cdot)\)
\(\chi_{1127}(573,\cdot)\)
...
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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 ]
\((346,442)\) → \((e\left(\frac{1}{14}\right),e\left(\frac{19}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1127 }(76, a) \) |
\(1\) | \(1\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{73}{154}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{97}{154}\right)\) | \(e\left(\frac{9}{154}\right)\) |
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sage:chi.jacobi_sum(n)
