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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(449, base_ring=CyclotomicField(224))
M = H._module
chi = DirichletCharacter(H, M([57]))
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pari:[g,chi] = znchar(Mod(154,449))
| Modulus: | \(449\) | |
| Conductor: | \(449\) |
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sage:chi.conductor()
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pari:znconreyconductor(g,chi)
|
| Order: | \(224\) |
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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)
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\(\chi_{449}(2,\cdot)\)
\(\chi_{449}(8,\cdot)\)
\(\chi_{449}(9,\cdot)\)
\(\chi_{449}(14,\cdot)\)
\(\chi_{449}(32,\cdot)\)
\(\chi_{449}(36,\cdot)\)
\(\chi_{449}(40,\cdot)\)
\(\chi_{449}(41,\cdot)\)
\(\chi_{449}(46,\cdot)\)
\(\chi_{449}(50,\cdot)\)
\(\chi_{449}(53,\cdot)\)
\(\chi_{449}(56,\cdot)\)
\(\chi_{449}(57,\cdot)\)
\(\chi_{449}(59,\cdot)\)
\(\chi_{449}(61,\cdot)\)
\(\chi_{449}(63,\cdot)\)
\(\chi_{449}(70,\cdot)\)
\(\chi_{449}(78,\cdot)\)
\(\chi_{449}(87,\cdot)\)
\(\chi_{449}(88,\cdot)\)
\(\chi_{449}(97,\cdot)\)
\(\chi_{449}(98,\cdot)\)
\(\chi_{449}(101,\cdot)\)
\(\chi_{449}(110,\cdot)\)
\(\chi_{449}(134,\cdot)\)
\(\chi_{449}(137,\cdot)\)
\(\chi_{449}(144,\cdot)\)
\(\chi_{449}(154,\cdot)\)
\(\chi_{449}(160,\cdot)\)
\(\chi_{449}(162,\cdot)\)
...
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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 ]
\(3\) → \(e\left(\frac{57}{224}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 449 }(154, a) \) |
\(1\) | \(1\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{57}{224}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{199}{224}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{28}\right)\) |
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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)
