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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2646, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([98,3]))
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gp:[g,chi] = znchar(Mod(1669, 2646))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2646.1669");
| Modulus: | \(2646\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1323\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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 \(\chi_{1323}(346,\cdot)\) |
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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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\(\chi_{2646}(61,\cdot)\)
\(\chi_{2646}(157,\cdot)\)
\(\chi_{2646}(187,\cdot)\)
\(\chi_{2646}(283,\cdot)\)
\(\chi_{2646}(409,\cdot)\)
\(\chi_{2646}(439,\cdot)\)
\(\chi_{2646}(535,\cdot)\)
\(\chi_{2646}(565,\cdot)\)
\(\chi_{2646}(661,\cdot)\)
\(\chi_{2646}(691,\cdot)\)
\(\chi_{2646}(787,\cdot)\)
\(\chi_{2646}(817,\cdot)\)
\(\chi_{2646}(943,\cdot)\)
\(\chi_{2646}(1039,\cdot)\)
\(\chi_{2646}(1069,\cdot)\)
\(\chi_{2646}(1165,\cdot)\)
\(\chi_{2646}(1291,\cdot)\)
\(\chi_{2646}(1321,\cdot)\)
\(\chi_{2646}(1417,\cdot)\)
\(\chi_{2646}(1447,\cdot)\)
\(\chi_{2646}(1543,\cdot)\)
\(\chi_{2646}(1573,\cdot)\)
\(\chi_{2646}(1669,\cdot)\)
\(\chi_{2646}(1699,\cdot)\)
\(\chi_{2646}(1825,\cdot)\)
\(\chi_{2646}(1921,\cdot)\)
\(\chi_{2646}(1951,\cdot)\)
\(\chi_{2646}(2047,\cdot)\)
\(\chi_{2646}(2173,\cdot)\)
\(\chi_{2646}(2203,\cdot)\)
...
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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: |
$\Q(\zeta_{63})$ |
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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 126 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((785,1081)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{1}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2646 }(1669, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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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)
