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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1392, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,14,1]))
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pari:[g,chi] = znchar(Mod(1133,1392))
| Modulus: | \(1392\) | |
| Conductor: | \(1392\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(28\) |
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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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\(\chi_{1392}(77,\cdot)\)
\(\chi_{1392}(677,\cdot)\)
\(\chi_{1392}(797,\cdot)\)
\(\chi_{1392}(917,\cdot)\)
\(\chi_{1392}(965,\cdot)\)
\(\chi_{1392}(989,\cdot)\)
\(\chi_{1392}(1013,\cdot)\)
\(\chi_{1392}(1133,\cdot)\)
\(\chi_{1392}(1157,\cdot)\)
\(\chi_{1392}(1181,\cdot)\)
\(\chi_{1392}(1229,\cdot)\)
\(\chi_{1392}(1349,\cdot)\)
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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 ]
\((175,1045,929,1249)\) → \((1,-i,-1,e\left(\frac{1}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 1392 }(1133, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) |
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sage:chi.jacobi_sum(n)
