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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(893, base_ring=CyclotomicField(414))
M = H._module
chi = DirichletCharacter(H, M([322,144]))
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gp:[g,chi] = znchar(Mod(158, 893))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("893.158");
| Modulus: | \(893\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(893\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(207\) |
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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: | yes |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{893}(4,\cdot)\)
\(\chi_{893}(6,\cdot)\)
\(\chi_{893}(9,\cdot)\)
\(\chi_{893}(16,\cdot)\)
\(\chi_{893}(17,\cdot)\)
\(\chi_{893}(24,\cdot)\)
\(\chi_{893}(25,\cdot)\)
\(\chi_{893}(28,\cdot)\)
\(\chi_{893}(36,\cdot)\)
\(\chi_{893}(42,\cdot)\)
\(\chi_{893}(54,\cdot)\)
\(\chi_{893}(55,\cdot)\)
\(\chi_{893}(61,\cdot)\)
\(\chi_{893}(63,\cdot)\)
\(\chi_{893}(74,\cdot)\)
\(\chi_{893}(81,\cdot)\)
\(\chi_{893}(100,\cdot)\)
\(\chi_{893}(101,\cdot)\)
\(\chi_{893}(111,\cdot)\)
\(\chi_{893}(112,\cdot)\)
\(\chi_{893}(118,\cdot)\)
\(\chi_{893}(119,\cdot)\)
\(\chi_{893}(130,\cdot)\)
\(\chi_{893}(131,\cdot)\)
\(\chi_{893}(149,\cdot)\)
\(\chi_{893}(150,\cdot)\)
\(\chi_{893}(157,\cdot)\)
\(\chi_{893}(158,\cdot)\)
\(\chi_{893}(168,\cdot)\)
\(\chi_{893}(169,\cdot)\)
...
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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 };
\((800,381)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{8}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 893 }(158, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{207}\right)\) | \(e\left(\frac{14}{207}\right)\) | \(e\left(\frac{16}{207}\right)\) | \(e\left(\frac{164}{207}\right)\) | \(e\left(\frac{22}{207}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{28}{207}\right)\) | \(e\left(\frac{172}{207}\right)\) | \(e\left(\frac{53}{69}\right)\) |
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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)
