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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8509, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,73]))
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gp:[g,chi] = znchar(Mod(4959, 8509))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8509.4959");
| Modulus: | \(8509\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(127\) |
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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 \(\chi_{127}(6,\cdot)\) |
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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);
|
\(\chi_{8509}(537,\cdot)\)
\(\chi_{8509}(604,\cdot)\)
\(\chi_{8509}(805,\cdot)\)
\(\chi_{8509}(872,\cdot)\)
\(\chi_{8509}(1073,\cdot)\)
\(\chi_{8509}(1475,\cdot)\)
\(\chi_{8509}(1609,\cdot)\)
\(\chi_{8509}(1743,\cdot)\)
\(\chi_{8509}(1944,\cdot)\)
\(\chi_{8509}(2011,\cdot)\)
\(\chi_{8509}(2078,\cdot)\)
\(\chi_{8509}(2212,\cdot)\)
\(\chi_{8509}(2480,\cdot)\)
\(\chi_{8509}(2547,\cdot)\)
\(\chi_{8509}(2681,\cdot)\)
\(\chi_{8509}(3284,\cdot)\)
\(\chi_{8509}(3418,\cdot)\)
\(\chi_{8509}(3485,\cdot)\)
\(\chi_{8509}(3686,\cdot)\)
\(\chi_{8509}(4155,\cdot)\)
\(\chi_{8509}(4490,\cdot)\)
\(\chi_{8509}(4557,\cdot)\)
\(\chi_{8509}(4959,\cdot)\)
\(\chi_{8509}(5562,\cdot)\)
\(\chi_{8509}(5763,\cdot)\)
\(\chi_{8509}(5897,\cdot)\)
\(\chi_{8509}(6433,\cdot)\)
\(\chi_{8509}(6500,\cdot)\)
\(\chi_{8509}(6701,\cdot)\)
\(\chi_{8509}(7103,\cdot)\)
...
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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 };
\((2414,3686)\) → \((1,e\left(\frac{73}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8509 }(4959, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{25}{63}\right)\) |
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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)
