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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6757, base_ring=CyclotomicField(1624))
M = H._module
chi = DirichletCharacter(H, M([464,469]))
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gp:[g,chi] = znchar(Mod(140, 6757))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6757.140");
| Modulus: | \(6757\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6757\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1624\) |
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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);
|
\(\chi_{6757}(20,\cdot)\)
\(\chi_{6757}(24,\cdot)\)
\(\chi_{6757}(45,\cdot)\)
\(\chi_{6757}(53,\cdot)\)
\(\chi_{6757}(54,\cdot)\)
\(\chi_{6757}(65,\cdot)\)
\(\chi_{6757}(78,\cdot)\)
\(\chi_{6757}(82,\cdot)\)
\(\chi_{6757}(83,\cdot)\)
\(\chi_{6757}(94,\cdot)\)
\(\chi_{6757}(103,\cdot)\)
\(\chi_{6757}(111,\cdot)\)
\(\chi_{6757}(139,\cdot)\)
\(\chi_{6757}(140,\cdot)\)
\(\chi_{6757}(165,\cdot)\)
\(\chi_{6757}(168,\cdot)\)
\(\chi_{6757}(190,\cdot)\)
\(\chi_{6757}(194,\cdot)\)
\(\chi_{6757}(198,\cdot)\)
\(\chi_{6757}(199,\cdot)\)
\(\chi_{6757}(223,\cdot)\)
\(\chi_{6757}(227,\cdot)\)
\(\chi_{6757}(228,\cdot)\)
\(\chi_{6757}(239,\cdot)\)
\(\chi_{6757}(255,\cdot)\)
\(\chi_{6757}(257,\cdot)\)
\(\chi_{6757}(268,\cdot)\)
\(\chi_{6757}(277,\cdot)\)
\(\chi_{6757}(281,\cdot)\)
\(\chi_{6757}(286,\cdot)\)
...
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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 };
\((234,6294)\) → \((e\left(\frac{2}{7}\right),e\left(\frac{67}{232}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 6757 }(140, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{16}{203}\right)\) | \(e\left(\frac{1165}{1624}\right)\) | \(e\left(\frac{32}{203}\right)\) | \(e\left(\frac{1521}{1624}\right)\) | \(e\left(\frac{1293}{1624}\right)\) | \(e\left(\frac{439}{812}\right)\) | \(e\left(\frac{48}{203}\right)\) | \(e\left(\frac{353}{812}\right)\) | \(e\left(\frac{25}{1624}\right)\) | \(e\left(\frac{57}{1624}\right)\) |
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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)
