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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,23]))
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gp:[g,chi] = znchar(Mod(41, 847))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.41");
| Modulus: | \(847\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(847\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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);
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\(\chi_{847}(6,\cdot)\)
\(\chi_{847}(13,\cdot)\)
\(\chi_{847}(41,\cdot)\)
\(\chi_{847}(62,\cdot)\)
\(\chi_{847}(83,\cdot)\)
\(\chi_{847}(90,\cdot)\)
\(\chi_{847}(139,\cdot)\)
\(\chi_{847}(160,\cdot)\)
\(\chi_{847}(167,\cdot)\)
\(\chi_{847}(195,\cdot)\)
\(\chi_{847}(216,\cdot)\)
\(\chi_{847}(237,\cdot)\)
\(\chi_{847}(244,\cdot)\)
\(\chi_{847}(272,\cdot)\)
\(\chi_{847}(293,\cdot)\)
\(\chi_{847}(314,\cdot)\)
\(\chi_{847}(321,\cdot)\)
\(\chi_{847}(349,\cdot)\)
\(\chi_{847}(370,\cdot)\)
\(\chi_{847}(391,\cdot)\)
\(\chi_{847}(398,\cdot)\)
\(\chi_{847}(426,\cdot)\)
\(\chi_{847}(447,\cdot)\)
\(\chi_{847}(468,\cdot)\)
\(\chi_{847}(503,\cdot)\)
\(\chi_{847}(545,\cdot)\)
\(\chi_{847}(552,\cdot)\)
\(\chi_{847}(580,\cdot)\)
\(\chi_{847}(601,\cdot)\)
\(\chi_{847}(622,\cdot)\)
...
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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 };
\((122,365)\) → \((-1,e\left(\frac{23}{110}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(41, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{34}{55}\right)\) |
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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)
