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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([14,7,14,10]))
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pari:[g,chi] = znchar(Mod(299,1392))
| Modulus: | \(1392\) | |
| Conductor: | \(1392\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
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| Order: | \(28\) |
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sage:chi.multiplicative_order()
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pari:charorder(g,chi)
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| 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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\(\chi_{1392}(35,\cdot)\)
\(\chi_{1392}(179,\cdot)\)
\(\chi_{1392}(299,\cdot)\)
\(\chi_{1392}(323,\cdot)\)
\(\chi_{1392}(419,\cdot)\)
\(\chi_{1392}(515,\cdot)\)
\(\chi_{1392}(731,\cdot)\)
\(\chi_{1392}(875,\cdot)\)
\(\chi_{1392}(995,\cdot)\)
\(\chi_{1392}(1019,\cdot)\)
\(\chi_{1392}(1115,\cdot)\)
\(\chi_{1392}(1211,\cdot)\)
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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 ]
\((175,1045,929,1249)\) → \((-1,i,-1,e\left(\frac{5}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 1392 }(299, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) |
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sage:chi.jacobi_sum(n)
