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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3325, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([12,10,15]))
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pari:[g,chi] = znchar(Mod(2431,3325))
| Modulus: | \(3325\) | |
| Conductor: | \(3325\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{3325}(436,\cdot)\)
\(\chi_{3325}(816,\cdot)\)
\(\chi_{3325}(1481,\cdot)\)
\(\chi_{3325}(1766,\cdot)\)
\(\chi_{3325}(2146,\cdot)\)
\(\chi_{3325}(2431,\cdot)\)
\(\chi_{3325}(2811,\cdot)\)
\(\chi_{3325}(3096,\cdot)\)
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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 ]
\((2927,2376,876)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{1}{3}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 3325 }(2431, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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sage:chi.jacobi_sum(n)
