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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1575, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,18,5]))
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pari:[g,chi] = znchar(Mod(1046,1575))
| Modulus: | \(1575\) | |
| Conductor: | \(1575\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| 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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\(\chi_{1575}(131,\cdot)\)
\(\chi_{1575}(416,\cdot)\)
\(\chi_{1575}(446,\cdot)\)
\(\chi_{1575}(731,\cdot)\)
\(\chi_{1575}(761,\cdot)\)
\(\chi_{1575}(1046,\cdot)\)
\(\chi_{1575}(1361,\cdot)\)
\(\chi_{1575}(1391,\cdot)\)
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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 ]
\((1226,127,451)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{5}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 1575 }(1046, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) |
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sage:chi.jacobi_sum(n)
