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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([29]))
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gp:[g,chi] = znchar(Mod(487, 722))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.487");
| Modulus: | \(722\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(361\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(114\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | no, induced from \(\chi_{361}(126,\cdot)\) |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{722}(27,\cdot)\)
\(\chi_{722}(31,\cdot)\)
\(\chi_{722}(65,\cdot)\)
\(\chi_{722}(103,\cdot)\)
\(\chi_{722}(107,\cdot)\)
\(\chi_{722}(141,\cdot)\)
\(\chi_{722}(145,\cdot)\)
\(\chi_{722}(179,\cdot)\)
\(\chi_{722}(183,\cdot)\)
\(\chi_{722}(217,\cdot)\)
\(\chi_{722}(221,\cdot)\)
\(\chi_{722}(255,\cdot)\)
\(\chi_{722}(259,\cdot)\)
\(\chi_{722}(297,\cdot)\)
\(\chi_{722}(331,\cdot)\)
\(\chi_{722}(335,\cdot)\)
\(\chi_{722}(369,\cdot)\)
\(\chi_{722}(373,\cdot)\)
\(\chi_{722}(407,\cdot)\)
\(\chi_{722}(411,\cdot)\)
\(\chi_{722}(445,\cdot)\)
\(\chi_{722}(449,\cdot)\)
\(\chi_{722}(483,\cdot)\)
\(\chi_{722}(487,\cdot)\)
\(\chi_{722}(521,\cdot)\)
\(\chi_{722}(525,\cdot)\)
\(\chi_{722}(559,\cdot)\)
\(\chi_{722}(563,\cdot)\)
\(\chi_{722}(597,\cdot)\)
\(\chi_{722}(601,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\(363\) → \(e\left(\frac{29}{114}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 722 }(487, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{8}{57}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
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sage:chi.gauss_sum(a)
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gp:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
