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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([0,125,126]))
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gp:[g,chi] = znchar(Mod(262, 6027))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6027.262");
| Modulus: | \(6027\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2009\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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 \(\chi_{2009}(262,\cdot)\) |
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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);
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\(\chi_{6027}(10,\cdot)\)
\(\chi_{6027}(262,\cdot)\)
\(\chi_{6027}(283,\cdot)\)
\(\chi_{6027}(346,\cdot)\)
\(\chi_{6027}(502,\cdot)\)
\(\chi_{6027}(775,\cdot)\)
\(\chi_{6027}(838,\cdot)\)
\(\chi_{6027}(871,\cdot)\)
\(\chi_{6027}(1123,\cdot)\)
\(\chi_{6027}(1144,\cdot)\)
\(\chi_{6027}(1363,\cdot)\)
\(\chi_{6027}(1615,\cdot)\)
\(\chi_{6027}(1699,\cdot)\)
\(\chi_{6027}(1732,\cdot)\)
\(\chi_{6027}(1984,\cdot)\)
\(\chi_{6027}(2005,\cdot)\)
\(\chi_{6027}(2068,\cdot)\)
\(\chi_{6027}(2476,\cdot)\)
\(\chi_{6027}(2497,\cdot)\)
\(\chi_{6027}(2560,\cdot)\)
\(\chi_{6027}(2593,\cdot)\)
\(\chi_{6027}(2845,\cdot)\)
\(\chi_{6027}(2866,\cdot)\)
\(\chi_{6027}(2929,\cdot)\)
\(\chi_{6027}(3085,\cdot)\)
\(\chi_{6027}(3337,\cdot)\)
\(\chi_{6027}(3358,\cdot)\)
\(\chi_{6027}(3421,\cdot)\)
\(\chi_{6027}(3454,\cdot)\)
\(\chi_{6027}(3727,\cdot)\)
...
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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 };
\((4019,493,2794)\) → \((1,e\left(\frac{25}{42}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 6027 }(262, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{16}{105}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{7}{30}\right)\) |
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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)
