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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([22,27]))
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pari:[g,chi] = znchar(Mod(347,405))
| Modulus: | \(405\) | |
| Conductor: | \(405\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(108\) |
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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: | even |
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sage:chi.is_odd()
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pari:zncharisodd(g,chi)
|
\(\chi_{405}(2,\cdot)\)
\(\chi_{405}(23,\cdot)\)
\(\chi_{405}(32,\cdot)\)
\(\chi_{405}(38,\cdot)\)
\(\chi_{405}(47,\cdot)\)
\(\chi_{405}(68,\cdot)\)
\(\chi_{405}(77,\cdot)\)
\(\chi_{405}(83,\cdot)\)
\(\chi_{405}(92,\cdot)\)
\(\chi_{405}(113,\cdot)\)
\(\chi_{405}(122,\cdot)\)
\(\chi_{405}(128,\cdot)\)
\(\chi_{405}(137,\cdot)\)
\(\chi_{405}(158,\cdot)\)
\(\chi_{405}(167,\cdot)\)
\(\chi_{405}(173,\cdot)\)
\(\chi_{405}(182,\cdot)\)
\(\chi_{405}(203,\cdot)\)
\(\chi_{405}(212,\cdot)\)
\(\chi_{405}(218,\cdot)\)
\(\chi_{405}(227,\cdot)\)
\(\chi_{405}(248,\cdot)\)
\(\chi_{405}(257,\cdot)\)
\(\chi_{405}(263,\cdot)\)
\(\chi_{405}(272,\cdot)\)
\(\chi_{405}(293,\cdot)\)
\(\chi_{405}(302,\cdot)\)
\(\chi_{405}(308,\cdot)\)
\(\chi_{405}(317,\cdot)\)
\(\chi_{405}(338,\cdot)\)
...
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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 ]
\((326,82)\) → \((e\left(\frac{11}{54}\right),i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 405 }(347, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
