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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1359, base_ring=CyclotomicField(150))
M = H._module
chi = DirichletCharacter(H, M([75,74]))
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gp:[g,chi] = znchar(Mod(629, 1359))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1359.629");
| Modulus: | \(1359\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(453\) |
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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 \(\chi_{453}(176,\cdot)\) |
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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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\(\chi_{1359}(17,\cdot)\)
\(\chi_{1359}(62,\cdot)\)
\(\chi_{1359}(80,\cdot)\)
\(\chi_{1359}(116,\cdot)\)
\(\chi_{1359}(161,\cdot)\)
\(\chi_{1359}(188,\cdot)\)
\(\chi_{1359}(206,\cdot)\)
\(\chi_{1359}(251,\cdot)\)
\(\chi_{1359}(287,\cdot)\)
\(\chi_{1359}(296,\cdot)\)
\(\chi_{1359}(323,\cdot)\)
\(\chi_{1359}(341,\cdot)\)
\(\chi_{1359}(440,\cdot)\)
\(\chi_{1359}(458,\cdot)\)
\(\chi_{1359}(548,\cdot)\)
\(\chi_{1359}(629,\cdot)\)
\(\chi_{1359}(638,\cdot)\)
\(\chi_{1359}(647,\cdot)\)
\(\chi_{1359}(692,\cdot)\)
\(\chi_{1359}(701,\cdot)\)
\(\chi_{1359}(773,\cdot)\)
\(\chi_{1359}(791,\cdot)\)
\(\chi_{1359}(800,\cdot)\)
\(\chi_{1359}(845,\cdot)\)
\(\chi_{1359}(854,\cdot)\)
\(\chi_{1359}(899,\cdot)\)
\(\chi_{1359}(917,\cdot)\)
\(\chi_{1359}(953,\cdot)\)
\(\chi_{1359}(980,\cdot)\)
\(\chi_{1359}(1043,\cdot)\)
...
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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()
|
\((605,1063)\) → \((-1,e\left(\frac{37}{75}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1359 }(629, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{2}{15}\right)\) |
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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)
