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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(831, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,4]))
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gp:[g,chi] = znchar(Mod(175, 831))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("831.175");
| Modulus: | \(831\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(277\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(23\) |
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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 \(\chi_{277}(175,\cdot)\) |
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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: | 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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\(\chi_{831}(16,\cdot)\)
\(\chi_{831}(19,\cdot)\)
\(\chi_{831}(52,\cdot)\)
\(\chi_{831}(157,\cdot)\)
\(\chi_{831}(169,\cdot)\)
\(\chi_{831}(175,\cdot)\)
\(\chi_{831}(211,\cdot)\)
\(\chi_{831}(256,\cdot)\)
\(\chi_{831}(304,\cdot)\)
\(\chi_{831}(307,\cdot)\)
\(\chi_{831}(346,\cdot)\)
\(\chi_{831}(361,\cdot)\)
\(\chi_{831}(478,\cdot)\)
\(\chi_{831}(490,\cdot)\)
\(\chi_{831}(541,\cdot)\)
\(\chi_{831}(550,\cdot)\)
\(\chi_{831}(685,\cdot)\)
\(\chi_{831}(709,\cdot)\)
\(\chi_{831}(718,\cdot)\)
\(\chi_{831}(757,\cdot)\)
\(\chi_{831}(772,\cdot)\)
\(\chi_{831}(790,\cdot)\)
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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 };
\((278,559)\) → \((1,e\left(\frac{2}{23}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 831 }(175, a) \) |
\(1\) | \(1\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) |
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sage:chi(x) # x integer
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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)
