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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(228672, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([132,99,132,206]))
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gp:[g,chi] = znchar(Mod(40823, 228672))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("228672.40823");
| Modulus: | \(228672\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(38112\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(264\) |
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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 \(\chi_{38112}(26531,\cdot)\) |
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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: | 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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\(\chi_{228672}(71,\cdot)\)
\(\chi_{228672}(6695,\cdot)\)
\(\chi_{228672}(7271,\cdot)\)
\(\chi_{228672}(13031,\cdot)\)
\(\chi_{228672}(15767,\cdot)\)
\(\chi_{228672}(17351,\cdot)\)
\(\chi_{228672}(19367,\cdot)\)
\(\chi_{228672}(21383,\cdot)\)
\(\chi_{228672}(22247,\cdot)\)
\(\chi_{228672}(33911,\cdot)\)
\(\chi_{228672}(34919,\cdot)\)
\(\chi_{228672}(36647,\cdot)\)
\(\chi_{228672}(37223,\cdot)\)
\(\chi_{228672}(40391,\cdot)\)
\(\chi_{228672}(40823,\cdot)\)
\(\chi_{228672}(42263,\cdot)\)
\(\chi_{228672}(42695,\cdot)\)
\(\chi_{228672}(44135,\cdot)\)
\(\chi_{228672}(44567,\cdot)\)
\(\chi_{228672}(47735,\cdot)\)
\(\chi_{228672}(48311,\cdot)\)
\(\chi_{228672}(50039,\cdot)\)
\(\chi_{228672}(51047,\cdot)\)
\(\chi_{228672}(62711,\cdot)\)
\(\chi_{228672}(63575,\cdot)\)
\(\chi_{228672}(65591,\cdot)\)
\(\chi_{228672}(67607,\cdot)\)
\(\chi_{228672}(69191,\cdot)\)
\(\chi_{228672}(71927,\cdot)\)
\(\chi_{228672}(77687,\cdot)\)
...
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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: |
$\Q(\zeta_{264})$ |
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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 264 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((21439,185797,25409,169921)\) → \((-1,e\left(\frac{3}{8}\right),-1,e\left(\frac{103}{132}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 228672 }(40823, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{173}{264}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{31}{264}\right)\) | \(e\left(\frac{115}{264}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{1}{264}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{85}{264}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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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)
