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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20800, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,35,48,60]))
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gp:[g,chi] = znchar(Mod(13421, 20800))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20800.13421");
| Modulus: | \(20800\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(20800\) |
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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: | 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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\(\chi_{20800}(21,\cdot)\)
\(\chi_{20800}(941,\cdot)\)
\(\chi_{20800}(1061,\cdot)\)
\(\chi_{20800}(1981,\cdot)\)
\(\chi_{20800}(3021,\cdot)\)
\(\chi_{20800}(3141,\cdot)\)
\(\chi_{20800}(4061,\cdot)\)
\(\chi_{20800}(4181,\cdot)\)
\(\chi_{20800}(5221,\cdot)\)
\(\chi_{20800}(6141,\cdot)\)
\(\chi_{20800}(6261,\cdot)\)
\(\chi_{20800}(7181,\cdot)\)
\(\chi_{20800}(8221,\cdot)\)
\(\chi_{20800}(8341,\cdot)\)
\(\chi_{20800}(9261,\cdot)\)
\(\chi_{20800}(9381,\cdot)\)
\(\chi_{20800}(10421,\cdot)\)
\(\chi_{20800}(11341,\cdot)\)
\(\chi_{20800}(11461,\cdot)\)
\(\chi_{20800}(12381,\cdot)\)
\(\chi_{20800}(13421,\cdot)\)
\(\chi_{20800}(13541,\cdot)\)
\(\chi_{20800}(14461,\cdot)\)
\(\chi_{20800}(14581,\cdot)\)
\(\chi_{20800}(15621,\cdot)\)
\(\chi_{20800}(16541,\cdot)\)
\(\chi_{20800}(16661,\cdot)\)
\(\chi_{20800}(17581,\cdot)\)
\(\chi_{20800}(18621,\cdot)\)
\(\chi_{20800}(18741,\cdot)\)
...
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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 };
\((12351,16901,14977,1601)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{3}{5}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 20800 }(13421, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) |
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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)
