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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([33,55,35]))
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pari:[g,chi] = znchar(Mod(1202,1407))
| 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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\(\chi_{1407}(80,\cdot)\)
\(\chi_{1407}(185,\cdot)\)
\(\chi_{1407}(299,\cdot)\)
\(\chi_{1407}(353,\cdot)\)
\(\chi_{1407}(584,\cdot)\)
\(\chi_{1407}(605,\cdot)\)
\(\chi_{1407}(614,\cdot)\)
\(\chi_{1407}(635,\cdot)\)
\(\chi_{1407}(647,\cdot)\)
\(\chi_{1407}(677,\cdot)\)
\(\chi_{1407}(731,\cdot)\)
\(\chi_{1407}(794,\cdot)\)
\(\chi_{1407}(824,\cdot)\)
\(\chi_{1407}(845,\cdot)\)
\(\chi_{1407}(899,\cdot)\)
\(\chi_{1407}(950,\cdot)\)
\(\chi_{1407}(1118,\cdot)\)
\(\chi_{1407}(1202,\cdot)\)
\(\chi_{1407}(1256,\cdot)\)
\(\chi_{1407}(1307,\cdot)\)
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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 ]
\((470,1207,337)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{35}{66}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1407 }(1202, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{31}{66}\right)\) |
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sage:chi.jacobi_sum(n)
