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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3775, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([0,127]))
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gp:[g,chi] = znchar(Mod(3076, 3775))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3775.3076");
| Modulus: | \(3775\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(151\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(150\) |
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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 \(\chi_{151}(56,\cdot)\) |
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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: | 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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\(\chi_{3775}(51,\cdot)\)
\(\chi_{3775}(126,\cdot)\)
\(\chi_{3775}(501,\cdot)\)
\(\chi_{3775}(826,\cdot)\)
\(\chi_{3775}(851,\cdot)\)
\(\chi_{3775}(901,\cdot)\)
\(\chi_{3775}(1026,\cdot)\)
\(\chi_{3775}(1301,\cdot)\)
\(\chi_{3775}(1476,\cdot)\)
\(\chi_{3775}(1651,\cdot)\)
\(\chi_{3775}(1676,\cdot)\)
\(\chi_{3775}(1776,\cdot)\)
\(\chi_{3775}(1801,\cdot)\)
\(\chi_{3775}(1826,\cdot)\)
\(\chi_{3775}(1901,\cdot)\)
\(\chi_{3775}(1926,\cdot)\)
\(\chi_{3775}(1976,\cdot)\)
\(\chi_{3775}(2026,\cdot)\)
\(\chi_{3775}(2126,\cdot)\)
\(\chi_{3775}(2226,\cdot)\)
\(\chi_{3775}(2326,\cdot)\)
\(\chi_{3775}(2376,\cdot)\)
\(\chi_{3775}(2451,\cdot)\)
\(\chi_{3775}(2676,\cdot)\)
\(\chi_{3775}(2701,\cdot)\)
\(\chi_{3775}(2826,\cdot)\)
\(\chi_{3775}(2851,\cdot)\)
\(\chi_{3775}(2876,\cdot)\)
\(\chi_{3775}(2951,\cdot)\)
\(\chi_{3775}(3026,\cdot)\)
...
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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: |
$\Q(\zeta_{75})$ |
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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 150 polynomial (not computed) |
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sage:chi.fixed_field()
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\((152,3026)\) → \((1,e\left(\frac{127}{150}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3775 }(3076, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{44}{75}\right)\) | \(e\left(\frac{17}{150}\right)\) | \(e\left(\frac{107}{150}\right)\) |
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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)
