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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2509, base_ring=CyclotomicField(192))
M = H._module
chi = DirichletCharacter(H, M([128,145]))
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gp:[g,chi] = znchar(Mod(984, 2509))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2509.984");
| Modulus: | \(2509\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2509\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(192\) |
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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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\(\chi_{2509}(22,\cdot)\)
\(\chi_{2509}(146,\cdot)\)
\(\chi_{2509}(152,\cdot)\)
\(\chi_{2509}(159,\cdot)\)
\(\chi_{2509}(178,\cdot)\)
\(\chi_{2509}(237,\cdot)\)
\(\chi_{2509}(308,\cdot)\)
\(\chi_{2509}(360,\cdot)\)
\(\chi_{2509}(412,\cdot)\)
\(\chi_{2509}(464,\cdot)\)
\(\chi_{2509}(497,\cdot)\)
\(\chi_{2509}(549,\cdot)\)
\(\chi_{2509}(594,\cdot)\)
\(\chi_{2509}(620,\cdot)\)
\(\chi_{2509}(640,\cdot)\)
\(\chi_{2509}(692,\cdot)\)
\(\chi_{2509}(750,\cdot)\)
\(\chi_{2509}(789,\cdot)\)
\(\chi_{2509}(809,\cdot)\)
\(\chi_{2509}(874,\cdot)\)
\(\chi_{2509}(913,\cdot)\)
\(\chi_{2509}(984,\cdot)\)
\(\chi_{2509}(1010,\cdot)\)
\(\chi_{2509}(1017,\cdot)\)
\(\chi_{2509}(1023,\cdot)\)
\(\chi_{2509}(1056,\cdot)\)
\(\chi_{2509}(1088,\cdot)\)
\(\chi_{2509}(1101,\cdot)\)
\(\chi_{2509}(1121,\cdot)\)
\(\chi_{2509}(1153,\cdot)\)
...
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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 };
\((1159,391)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{145}{192}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2509 }(984, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{145}{192}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{192}\right)\) | \(e\left(\frac{167}{192}\right)\) |
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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)
