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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20800, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([0,195,144,160]))
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gp:[g,chi] = znchar(Mod(1621, 20800))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20800.1621");
| Modulus: | \(20800\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(20800\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(240\) |
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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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\(\chi_{20800}(61,\cdot)\)
\(\chi_{20800}(341,\cdot)\)
\(\chi_{20800}(581,\cdot)\)
\(\chi_{20800}(861,\cdot)\)
\(\chi_{20800}(1381,\cdot)\)
\(\chi_{20800}(1621,\cdot)\)
\(\chi_{20800}(2141,\cdot)\)
\(\chi_{20800}(2421,\cdot)\)
\(\chi_{20800}(2661,\cdot)\)
\(\chi_{20800}(2941,\cdot)\)
\(\chi_{20800}(3181,\cdot)\)
\(\chi_{20800}(3461,\cdot)\)
\(\chi_{20800}(3981,\cdot)\)
\(\chi_{20800}(4221,\cdot)\)
\(\chi_{20800}(4741,\cdot)\)
\(\chi_{20800}(5021,\cdot)\)
\(\chi_{20800}(5261,\cdot)\)
\(\chi_{20800}(5541,\cdot)\)
\(\chi_{20800}(5781,\cdot)\)
\(\chi_{20800}(6061,\cdot)\)
\(\chi_{20800}(6581,\cdot)\)
\(\chi_{20800}(6821,\cdot)\)
\(\chi_{20800}(7341,\cdot)\)
\(\chi_{20800}(7621,\cdot)\)
\(\chi_{20800}(7861,\cdot)\)
\(\chi_{20800}(8141,\cdot)\)
\(\chi_{20800}(8381,\cdot)\)
\(\chi_{20800}(8661,\cdot)\)
\(\chi_{20800}(9181,\cdot)\)
\(\chi_{20800}(9421,\cdot)\)
...
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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: |
$\Q(\zeta_{240})$ |
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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 240 polynomial (not computed) |
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sage:chi.fixed_field()
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\((12351,16901,14977,1601)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{3}{5}\right),e\left(\frac{2}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 20800 }(1621, a) \) |
\(1\) | \(1\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{79}{240}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{193}{240}\right)\) |
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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)
