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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38407, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([187,228]))
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gp:[g,chi] = znchar(Mod(2525, 38407))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38407.2525");
| Modulus: | \(38407\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(38407\) |
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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: | 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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{38407}(138,\cdot)\)
\(\chi_{38407}(181,\cdot)\)
\(\chi_{38407}(977,\cdot)\)
\(\chi_{38407}(1489,\cdot)\)
\(\chi_{38407}(1730,\cdot)\)
\(\chi_{38407}(2068,\cdot)\)
\(\chi_{38407}(2525,\cdot)\)
\(\chi_{38407}(2647,\cdot)\)
\(\chi_{38407}(3081,\cdot)\)
\(\chi_{38407}(3269,\cdot)\)
\(\chi_{38407}(3660,\cdot)\)
\(\chi_{38407}(4065,\cdot)\)
\(\chi_{38407}(4239,\cdot)\)
\(\chi_{38407}(4253,\cdot)\)
\(\chi_{38407}(5156,\cdot)\)
\(\chi_{38407}(5845,\cdot)\)
\(\chi_{38407}(6578,\cdot)\)
\(\chi_{38407}(6748,\cdot)\)
\(\chi_{38407}(7897,\cdot)\)
\(\chi_{38407}(8244,\cdot)\)
\(\chi_{38407}(8894,\cdot)\)
\(\chi_{38407}(9836,\cdot)\)
\(\chi_{38407}(10024,\cdot)\)
\(\chi_{38407}(10245,\cdot)\)
\(\chi_{38407}(10820,\cdot)\)
\(\chi_{38407}(10824,\cdot)\)
\(\chi_{38407}(11403,\cdot)\)
\(\chi_{38407}(13912,\cdot)\)
\(\chi_{38407}(14266,\cdot)\)
\(\chi_{38407}(14999,\cdot)\)
...
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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 };
\((12936,12739)\) → \((e\left(\frac{17}{24}\right),e\left(\frac{19}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 38407 }(2525, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{235}{264}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{137}{264}\right)\) | \(e\left(\frac{75}{88}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
