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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19200, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,75,40,4]))
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gp:[g,chi] = znchar(Mod(2927, 19200))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19200.2927");
| Modulus: | \(19200\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4800\) |
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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: | no, induced from \(\chi_{4800}(3827,\cdot)\) |
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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: | no |
| 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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\(\chi_{19200}(47,\cdot)\)
\(\chi_{19200}(1103,\cdot)\)
\(\chi_{19200}(1967,\cdot)\)
\(\chi_{19200}(2063,\cdot)\)
\(\chi_{19200}(2927,\cdot)\)
\(\chi_{19200}(3023,\cdot)\)
\(\chi_{19200}(3887,\cdot)\)
\(\chi_{19200}(3983,\cdot)\)
\(\chi_{19200}(4847,\cdot)\)
\(\chi_{19200}(5903,\cdot)\)
\(\chi_{19200}(6767,\cdot)\)
\(\chi_{19200}(6863,\cdot)\)
\(\chi_{19200}(7727,\cdot)\)
\(\chi_{19200}(7823,\cdot)\)
\(\chi_{19200}(8687,\cdot)\)
\(\chi_{19200}(8783,\cdot)\)
\(\chi_{19200}(9647,\cdot)\)
\(\chi_{19200}(10703,\cdot)\)
\(\chi_{19200}(11567,\cdot)\)
\(\chi_{19200}(11663,\cdot)\)
\(\chi_{19200}(12527,\cdot)\)
\(\chi_{19200}(12623,\cdot)\)
\(\chi_{19200}(13487,\cdot)\)
\(\chi_{19200}(13583,\cdot)\)
\(\chi_{19200}(14447,\cdot)\)
\(\chi_{19200}(15503,\cdot)\)
\(\chi_{19200}(16367,\cdot)\)
\(\chi_{19200}(16463,\cdot)\)
\(\chi_{19200}(17327,\cdot)\)
\(\chi_{19200}(17423,\cdot)\)
...
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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 };
\((4351,10501,6401,5377)\) → \((-1,e\left(\frac{15}{16}\right),-1,e\left(\frac{1}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 19200 }(2927, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) |
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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)
