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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,14,13]))
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pari:[g,chi] = znchar(Mod(1039,1764))
| Modulus: | \(1764\) | |
| Conductor: | \(1764\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(42\) |
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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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\(\chi_{1764}(187,\cdot)\)
\(\chi_{1764}(283,\cdot)\)
\(\chi_{1764}(439,\cdot)\)
\(\chi_{1764}(535,\cdot)\)
\(\chi_{1764}(691,\cdot)\)
\(\chi_{1764}(787,\cdot)\)
\(\chi_{1764}(943,\cdot)\)
\(\chi_{1764}(1039,\cdot)\)
\(\chi_{1764}(1291,\cdot)\)
\(\chi_{1764}(1447,\cdot)\)
\(\chi_{1764}(1543,\cdot)\)
\(\chi_{1764}(1699,\cdot)\)
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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 ]
\((883,785,1081)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{13}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1764 }(1039, a) \) |
\(1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{21}\right)\) |
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sage:chi.jacobi_sum(n)
