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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84100, base_ring=CyclotomicField(406))
M = H._module
chi = DirichletCharacter(H, M([0,203,220]))
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gp:[g,chi] = znchar(Mod(5649, 84100))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("84100.5649");
| Modulus: | \(84100\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4205\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(406\) |
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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 \(\chi_{4205}(1444,\cdot)\) |
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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: | 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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\(\chi_{84100}(49,\cdot)\)
\(\chi_{84100}(749,\cdot)\)
\(\chi_{84100}(1649,\cdot)\)
\(\chi_{84100}(2249,\cdot)\)
\(\chi_{84100}(2749,\cdot)\)
\(\chi_{84100}(2849,\cdot)\)
\(\chi_{84100}(2949,\cdot)\)
\(\chi_{84100}(3649,\cdot)\)
\(\chi_{84100}(4549,\cdot)\)
\(\chi_{84100}(5149,\cdot)\)
\(\chi_{84100}(5649,\cdot)\)
\(\chi_{84100}(5749,\cdot)\)
\(\chi_{84100}(5849,\cdot)\)
\(\chi_{84100}(6549,\cdot)\)
\(\chi_{84100}(7449,\cdot)\)
\(\chi_{84100}(8049,\cdot)\)
\(\chi_{84100}(8549,\cdot)\)
\(\chi_{84100}(8649,\cdot)\)
\(\chi_{84100}(8749,\cdot)\)
\(\chi_{84100}(9449,\cdot)\)
\(\chi_{84100}(10349,\cdot)\)
\(\chi_{84100}(10949,\cdot)\)
\(\chi_{84100}(11449,\cdot)\)
\(\chi_{84100}(11549,\cdot)\)
\(\chi_{84100}(11649,\cdot)\)
\(\chi_{84100}(12349,\cdot)\)
\(\chi_{84100}(13249,\cdot)\)
\(\chi_{84100}(13849,\cdot)\)
\(\chi_{84100}(14349,\cdot)\)
\(\chi_{84100}(14449,\cdot)\)
...
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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 };
\((42051,30277,32801)\) → \((1,-1,e\left(\frac{110}{203}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 84100 }(5649, a) \) |
\(1\) | \(1\) | \(e\left(\frac{197}{406}\right)\) | \(e\left(\frac{99}{406}\right)\) | \(e\left(\frac{197}{203}\right)\) | \(e\left(\frac{181}{203}\right)\) | \(e\left(\frac{89}{406}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{136}{203}\right)\) | \(e\left(\frac{148}{203}\right)\) | \(e\left(\frac{67}{406}\right)\) | \(e\left(\frac{185}{406}\right)\) |
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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)
