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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3267, base_ring=CyclotomicField(990))
M = H._module
chi = DirichletCharacter(H, M([770,108]))
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gp:[g,chi] = znchar(Mod(103, 3267))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3267.103");
| Modulus: | \(3267\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3267\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(495\) |
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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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\(\chi_{3267}(4,\cdot)\)
\(\chi_{3267}(16,\cdot)\)
\(\chi_{3267}(25,\cdot)\)
\(\chi_{3267}(31,\cdot)\)
\(\chi_{3267}(49,\cdot)\)
\(\chi_{3267}(58,\cdot)\)
\(\chi_{3267}(70,\cdot)\)
\(\chi_{3267}(97,\cdot)\)
\(\chi_{3267}(103,\cdot)\)
\(\chi_{3267}(115,\cdot)\)
\(\chi_{3267}(157,\cdot)\)
\(\chi_{3267}(169,\cdot)\)
\(\chi_{3267}(196,\cdot)\)
\(\chi_{3267}(214,\cdot)\)
\(\chi_{3267}(223,\cdot)\)
\(\chi_{3267}(229,\cdot)\)
\(\chi_{3267}(247,\cdot)\)
\(\chi_{3267}(256,\cdot)\)
\(\chi_{3267}(268,\cdot)\)
\(\chi_{3267}(295,\cdot)\)
\(\chi_{3267}(301,\cdot)\)
\(\chi_{3267}(313,\cdot)\)
\(\chi_{3267}(322,\cdot)\)
\(\chi_{3267}(328,\cdot)\)
\(\chi_{3267}(346,\cdot)\)
\(\chi_{3267}(355,\cdot)\)
\(\chi_{3267}(367,\cdot)\)
\(\chi_{3267}(394,\cdot)\)
\(\chi_{3267}(400,\cdot)\)
\(\chi_{3267}(412,\cdot)\)
...
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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 };
| Field of values: |
$\Q(\zeta_{495})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 495 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((3026,244)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{6}{55}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 3267 }(103, a) \) |
\(1\) | \(1\) | \(e\left(\frac{439}{495}\right)\) | \(e\left(\frac{383}{495}\right)\) | \(e\left(\frac{476}{495}\right)\) | \(e\left(\frac{103}{495}\right)\) | \(e\left(\frac{109}{165}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{119}{495}\right)\) | \(e\left(\frac{47}{495}\right)\) | \(e\left(\frac{271}{495}\right)\) | \(e\left(\frac{2}{165}\right)\) |
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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)
