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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,16]))
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pari:[g,chi] = znchar(Mod(1283,1407))
| Modulus: | \(1407\) | |
| Conductor: | \(1407\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(66\) |
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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
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1407}(23,\cdot)\)
\(\chi_{1407}(65,\cdot)\)
\(\chi_{1407}(86,\cdot)\)
\(\chi_{1407}(317,\cdot)\)
\(\chi_{1407}(368,\cdot)\)
\(\chi_{1407}(473,\cdot)\)
\(\chi_{1407}(485,\cdot)\)
\(\chi_{1407}(557,\cdot)\)
\(\chi_{1407}(590,\cdot)\)
\(\chi_{1407}(725,\cdot)\)
\(\chi_{1407}(821,\cdot)\)
\(\chi_{1407}(830,\cdot)\)
\(\chi_{1407}(851,\cdot)\)
\(\chi_{1407}(998,\cdot)\)
\(\chi_{1407}(1040,\cdot)\)
\(\chi_{1407}(1061,\cdot)\)
\(\chi_{1407}(1178,\cdot)\)
\(\chi_{1407}(1283,\cdot)\)
\(\chi_{1407}(1346,\cdot)\)
\(\chi_{1407}(1376,\cdot)\)
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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 ]
\((470,1207,337)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{8}{33}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1407 }(1283, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) |
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sage:chi.jacobi_sum(n)
