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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7175, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([36,100,99]))
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gp:[g,chi] = znchar(Mod(4814, 7175))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7175.4814");
| Modulus: | \(7175\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(7175\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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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\(\chi_{7175}(19,\cdot)\)
\(\chi_{7175}(54,\cdot)\)
\(\chi_{7175}(129,\cdot)\)
\(\chi_{7175}(234,\cdot)\)
\(\chi_{7175}(339,\cdot)\)
\(\chi_{7175}(404,\cdot)\)
\(\chi_{7175}(444,\cdot)\)
\(\chi_{7175}(479,\cdot)\)
\(\chi_{7175}(719,\cdot)\)
\(\chi_{7175}(1319,\cdot)\)
\(\chi_{7175}(1664,\cdot)\)
\(\chi_{7175}(1734,\cdot)\)
\(\chi_{7175}(1739,\cdot)\)
\(\chi_{7175}(2434,\cdot)\)
\(\chi_{7175}(2609,\cdot)\)
\(\chi_{7175}(3064,\cdot)\)
\(\chi_{7175}(3204,\cdot)\)
\(\chi_{7175}(3309,\cdot)\)
\(\chi_{7175}(3414,\cdot)\)
\(\chi_{7175}(3519,\cdot)\)
\(\chi_{7175}(3554,\cdot)\)
\(\chi_{7175}(4119,\cdot)\)
\(\chi_{7175}(4154,\cdot)\)
\(\chi_{7175}(4394,\cdot)\)
\(\chi_{7175}(4504,\cdot)\)
\(\chi_{7175}(4814,\cdot)\)
\(\chi_{7175}(4819,\cdot)\)
\(\chi_{7175}(5764,\cdot)\)
\(\chi_{7175}(5834,\cdot)\)
\(\chi_{7175}(6534,\cdot)\)
...
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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 };
\((6602,3076,5951)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{5}{6}\right),e\left(\frac{33}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 7175 }(4814, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(i\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{2}{3}\right)\) |
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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)
