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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([40,65,52]))
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gp:[g,chi] = znchar(Mod(1067, 1600))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1600.1067");
| 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);
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\(\chi_{1600}(3,\cdot)\)
\(\chi_{1600}(27,\cdot)\)
\(\chi_{1600}(83,\cdot)\)
\(\chi_{1600}(163,\cdot)\)
\(\chi_{1600}(187,\cdot)\)
\(\chi_{1600}(267,\cdot)\)
\(\chi_{1600}(323,\cdot)\)
\(\chi_{1600}(347,\cdot)\)
\(\chi_{1600}(403,\cdot)\)
\(\chi_{1600}(427,\cdot)\)
\(\chi_{1600}(483,\cdot)\)
\(\chi_{1600}(563,\cdot)\)
\(\chi_{1600}(587,\cdot)\)
\(\chi_{1600}(667,\cdot)\)
\(\chi_{1600}(723,\cdot)\)
\(\chi_{1600}(747,\cdot)\)
\(\chi_{1600}(803,\cdot)\)
\(\chi_{1600}(827,\cdot)\)
\(\chi_{1600}(883,\cdot)\)
\(\chi_{1600}(963,\cdot)\)
\(\chi_{1600}(987,\cdot)\)
\(\chi_{1600}(1067,\cdot)\)
\(\chi_{1600}(1123,\cdot)\)
\(\chi_{1600}(1147,\cdot)\)
\(\chi_{1600}(1203,\cdot)\)
\(\chi_{1600}(1227,\cdot)\)
\(\chi_{1600}(1283,\cdot)\)
\(\chi_{1600}(1363,\cdot)\)
\(\chi_{1600}(1387,\cdot)\)
\(\chi_{1600}(1467,\cdot)\)
...
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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 };
\((1151,901,577)\) → \((-1,e\left(\frac{13}{16}\right),e\left(\frac{13}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1600 }(1067, a) \) |
\(1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{37}{80}\right)\) |
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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)
