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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,21,20]))
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pari:[g,chi] = znchar(Mod(1859,2100))
| Modulus: | \(2100\) | |
| Conductor: | \(2100\) |
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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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\(\chi_{2100}(179,\cdot)\)
\(\chi_{2100}(359,\cdot)\)
\(\chi_{2100}(779,\cdot)\)
\(\chi_{2100}(1019,\cdot)\)
\(\chi_{2100}(1439,\cdot)\)
\(\chi_{2100}(1619,\cdot)\)
\(\chi_{2100}(1859,\cdot)\)
\(\chi_{2100}(2039,\cdot)\)
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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 ]
\((1051,701,1177,1501)\) → \((-1,-1,e\left(\frac{7}{10}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 2100 }(1859, a) \) |
\(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(1\) |
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sage:chi.jacobi_sum(n)
