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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29070, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,108,117,64]))
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gp:[g,chi] = znchar(Mod(6013, 29070))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29070.6013");
| Modulus: | \(29070\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1615\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(144\) |
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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 \(\chi_{1615}(1168,\cdot)\) |
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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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\(\chi_{29070}(73,\cdot)\)
\(\chi_{29070}(397,\cdot)\)
\(\chi_{29070}(1423,\cdot)\)
\(\chi_{29070}(2683,\cdot)\)
\(\chi_{29070}(3133,\cdot)\)
\(\chi_{29070}(4987,\cdot)\)
\(\chi_{29070}(5077,\cdot)\)
\(\chi_{29070}(5743,\cdot)\)
\(\chi_{29070}(6013,\cdot)\)
\(\chi_{29070}(7723,\cdot)\)
\(\chi_{29070}(8137,\cdot)\)
\(\chi_{29070}(8803,\cdot)\)
\(\chi_{29070}(9523,\cdot)\)
\(\chi_{29070}(10207,\cdot)\)
\(\chi_{29070}(10333,\cdot)\)
\(\chi_{29070}(11197,\cdot)\)
\(\chi_{29070}(12583,\cdot)\)
\(\chi_{29070}(12727,\cdot)\)
\(\chi_{29070}(13267,\cdot)\)
\(\chi_{29070}(13393,\cdot)\)
\(\chi_{29070}(13627,\cdot)\)
\(\chi_{29070}(15643,\cdot)\)
\(\chi_{29070}(15787,\cdot)\)
\(\chi_{29070}(16327,\cdot)\)
\(\chi_{29070}(16687,\cdot)\)
\(\chi_{29070}(17173,\cdot)\)
\(\chi_{29070}(17857,\cdot)\)
\(\chi_{29070}(17983,\cdot)\)
\(\chi_{29070}(18757,\cdot)\)
\(\chi_{29070}(19747,\cdot)\)
...
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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 };
\((25841,23257,11971,3061)\) → \((1,-i,e\left(\frac{13}{16}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 29070 }(6013, a) \) |
\(1\) | \(1\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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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)
