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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1275, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,12,35]))
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pari:[g,chi] = znchar(Mod(614,1275))
| Modulus: | \(1275\) | |
| Conductor: | \(1275\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(40\) |
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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: | odd |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
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\(\chi_{1275}(59,\cdot)\)
\(\chi_{1275}(104,\cdot)\)
\(\chi_{1275}(134,\cdot)\)
\(\chi_{1275}(179,\cdot)\)
\(\chi_{1275}(314,\cdot)\)
\(\chi_{1275}(359,\cdot)\)
\(\chi_{1275}(389,\cdot)\)
\(\chi_{1275}(434,\cdot)\)
\(\chi_{1275}(569,\cdot)\)
\(\chi_{1275}(614,\cdot)\)
\(\chi_{1275}(644,\cdot)\)
\(\chi_{1275}(689,\cdot)\)
\(\chi_{1275}(869,\cdot)\)
\(\chi_{1275}(944,\cdot)\)
\(\chi_{1275}(1079,\cdot)\)
\(\chi_{1275}(1154,\cdot)\)
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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 ]
\((851,52,751)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{7}{8}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(19\) | \(22\) |
| \( \chi_{ 1275 }(614, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) |
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sage:chi.jacobi_sum(n)
