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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1588, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([66,103]))
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pari:[g,chi] = znchar(Mod(1123,1588))
| Modulus: | \(1588\) | |
| Conductor: | \(1588\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(132\) |
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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)
|
\(\chi_{1588}(15,\cdot)\)
\(\chi_{1588}(55,\cdot)\)
\(\chi_{1588}(71,\cdot)\)
\(\chi_{1588}(95,\cdot)\)
\(\chi_{1588}(103,\cdot)\)
\(\chi_{1588}(123,\cdot)\)
\(\chi_{1588}(231,\cdot)\)
\(\chi_{1588}(267,\cdot)\)
\(\chi_{1588}(311,\cdot)\)
\(\chi_{1588}(327,\cdot)\)
\(\chi_{1588}(343,\cdot)\)
\(\chi_{1588}(451,\cdot)\)
\(\chi_{1588}(467,\cdot)\)
\(\chi_{1588}(483,\cdot)\)
\(\chi_{1588}(527,\cdot)\)
\(\chi_{1588}(563,\cdot)\)
\(\chi_{1588}(671,\cdot)\)
\(\chi_{1588}(691,\cdot)\)
\(\chi_{1588}(699,\cdot)\)
\(\chi_{1588}(723,\cdot)\)
\(\chi_{1588}(739,\cdot)\)
\(\chi_{1588}(779,\cdot)\)
\(\chi_{1588}(811,\cdot)\)
\(\chi_{1588}(847,\cdot)\)
\(\chi_{1588}(907,\cdot)\)
\(\chi_{1588}(911,\cdot)\)
\(\chi_{1588}(919,\cdot)\)
\(\chi_{1588}(975,\cdot)\)
\(\chi_{1588}(979,\cdot)\)
\(\chi_{1588}(1123,\cdot)\)
...
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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 ]
\((795,5)\) → \((-1,e\left(\frac{103}{132}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1588 }(1123, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{97}{132}\right)\) |
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sage:chi.jacobi_sum(n)
