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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3825, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([10,27,15]))
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pari:[g,chi] = znchar(Mod(1444,3825))
| Modulus: | \(3825\) | |
| Conductor: | \(3825\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(30\) |
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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
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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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\(\chi_{3825}(169,\cdot)\)
\(\chi_{3825}(679,\cdot)\)
\(\chi_{3825}(934,\cdot)\)
\(\chi_{3825}(1444,\cdot)\)
\(\chi_{3825}(2209,\cdot)\)
\(\chi_{3825}(2464,\cdot)\)
\(\chi_{3825}(3229,\cdot)\)
\(\chi_{3825}(3739,\cdot)\)
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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 ]
\((2126,2602,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{10}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 3825 }(1444, a) \) |
\(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) |
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sage:chi.jacobi_sum(n)
