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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,35,48]))
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gp:[g,chi] = znchar(Mod(621, 1600))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1600.621");
| Modulus: | \(1600\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1600\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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);
|
\(\chi_{1600}(21,\cdot)\)
\(\chi_{1600}(61,\cdot)\)
\(\chi_{1600}(141,\cdot)\)
\(\chi_{1600}(181,\cdot)\)
\(\chi_{1600}(221,\cdot)\)
\(\chi_{1600}(261,\cdot)\)
\(\chi_{1600}(341,\cdot)\)
\(\chi_{1600}(381,\cdot)\)
\(\chi_{1600}(421,\cdot)\)
\(\chi_{1600}(461,\cdot)\)
\(\chi_{1600}(541,\cdot)\)
\(\chi_{1600}(581,\cdot)\)
\(\chi_{1600}(621,\cdot)\)
\(\chi_{1600}(661,\cdot)\)
\(\chi_{1600}(741,\cdot)\)
\(\chi_{1600}(781,\cdot)\)
\(\chi_{1600}(821,\cdot)\)
\(\chi_{1600}(861,\cdot)\)
\(\chi_{1600}(941,\cdot)\)
\(\chi_{1600}(981,\cdot)\)
\(\chi_{1600}(1021,\cdot)\)
\(\chi_{1600}(1061,\cdot)\)
\(\chi_{1600}(1141,\cdot)\)
\(\chi_{1600}(1181,\cdot)\)
\(\chi_{1600}(1221,\cdot)\)
\(\chi_{1600}(1261,\cdot)\)
\(\chi_{1600}(1341,\cdot)\)
\(\chi_{1600}(1381,\cdot)\)
\(\chi_{1600}(1421,\cdot)\)
\(\chi_{1600}(1461,\cdot)\)
...
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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 };
\((1151,901,577)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1600 }(621, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) |
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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)
