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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(549261, base_ring=CyclotomicField(3390))
M = H._module
chi = DirichletCharacter(H, M([2825,412]))
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gp:[g,chi] = znchar(Mod(1673, 549261))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("549261.1673");
| Modulus: | \(549261\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(61029\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(3390\) |
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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 \(\chi_{61029}(42359,\cdot)\) |
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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);
|
\(\chi_{549261}(26,\cdot)\)
\(\chi_{549261}(539,\cdot)\)
\(\chi_{549261}(1079,\cdot)\)
\(\chi_{549261}(1673,\cdot)\)
\(\chi_{549261}(2537,\cdot)\)
\(\chi_{549261}(2618,\cdot)\)
\(\chi_{549261}(2645,\cdot)\)
\(\chi_{549261}(3050,\cdot)\)
\(\chi_{549261}(4319,\cdot)\)
\(\chi_{549261}(4346,\cdot)\)
\(\chi_{549261}(5777,\cdot)\)
\(\chi_{549261}(6020,\cdot)\)
\(\chi_{549261}(6668,\cdot)\)
\(\chi_{549261}(8126,\cdot)\)
\(\chi_{549261}(9206,\cdot)\)
\(\chi_{549261}(9692,\cdot)\)
\(\chi_{549261}(10583,\cdot)\)
\(\chi_{549261}(11042,\cdot)\)
\(\chi_{549261}(11879,\cdot)\)
\(\chi_{549261}(12500,\cdot)\)
\(\chi_{549261}(12905,\cdot)\)
\(\chi_{549261}(12986,\cdot)\)
\(\chi_{549261}(13958,\cdot)\)
\(\chi_{549261}(15173,\cdot)\)
\(\chi_{549261}(15281,\cdot)\)
\(\chi_{549261}(15929,\cdot)\)
\(\chi_{549261}(16010,\cdot)\)
\(\chi_{549261}(17603,\cdot)\)
\(\chi_{549261}(17873,\cdot)\)
\(\chi_{549261}(18089,\cdot)\)
...
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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: |
$\Q(\zeta_{1695})$ |
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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 3390 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((47468,501796)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{206}{1695}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 549261 }(1673, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1079}{1130}\right)\) | \(e\left(\frac{514}{565}\right)\) | \(e\left(\frac{2063}{3390}\right)\) | \(e\left(\frac{212}{1695}\right)\) | \(e\left(\frac{977}{1130}\right)\) | \(e\left(\frac{191}{339}\right)\) | \(e\left(\frac{1103}{1130}\right)\) | \(e\left(\frac{1523}{1695}\right)\) | \(e\left(\frac{271}{3390}\right)\) | \(e\left(\frac{463}{565}\right)\) |
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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)
