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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12575, base_ring=CyclotomicField(1004))
M = H._module
chi = DirichletCharacter(H, M([753,992]))
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gp:[g,chi] = znchar(Mod(393, 12575))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12575.393");
| Modulus: | \(12575\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2515\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1004\) |
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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 \(\chi_{2515}(393,\cdot)\) |
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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);
|
\(\chi_{12575}(7,\cdot)\)
\(\chi_{12575}(18,\cdot)\)
\(\chi_{12575}(32,\cdot)\)
\(\chi_{12575}(43,\cdot)\)
\(\chi_{12575}(118,\cdot)\)
\(\chi_{12575}(132,\cdot)\)
\(\chi_{12575}(143,\cdot)\)
\(\chi_{12575}(168,\cdot)\)
\(\chi_{12575}(182,\cdot)\)
\(\chi_{12575}(207,\cdot)\)
\(\chi_{12575}(243,\cdot)\)
\(\chi_{12575}(257,\cdot)\)
\(\chi_{12575}(268,\cdot)\)
\(\chi_{12575}(282,\cdot)\)
\(\chi_{12575}(293,\cdot)\)
\(\chi_{12575}(332,\cdot)\)
\(\chi_{12575}(343,\cdot)\)
\(\chi_{12575}(368,\cdot)\)
\(\chi_{12575}(393,\cdot)\)
\(\chi_{12575}(432,\cdot)\)
\(\chi_{12575}(443,\cdot)\)
\(\chi_{12575}(468,\cdot)\)
\(\chi_{12575}(493,\cdot)\)
\(\chi_{12575}(507,\cdot)\)
\(\chi_{12575}(557,\cdot)\)
\(\chi_{12575}(582,\cdot)\)
\(\chi_{12575}(607,\cdot)\)
\(\chi_{12575}(632,\cdot)\)
\(\chi_{12575}(657,\cdot)\)
\(\chi_{12575}(693,\cdot)\)
...
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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 };
\((8552,3526)\) → \((-i,e\left(\frac{248}{251}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 12575 }(393, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{337}{1004}\right)\) | \(e\left(\frac{387}{1004}\right)\) | \(e\left(\frac{337}{502}\right)\) | \(e\left(\frac{181}{251}\right)\) | \(e\left(\frac{725}{1004}\right)\) | \(e\left(\frac{7}{1004}\right)\) | \(e\left(\frac{387}{502}\right)\) | \(e\left(\frac{125}{251}\right)\) | \(e\left(\frac{57}{1004}\right)\) | \(e\left(\frac{771}{1004}\right)\) |
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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)
