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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8649, base_ring=CyclotomicField(930))
M = H._module
chi = DirichletCharacter(H, M([155,42]))
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gp:[g,chi] = znchar(Mod(380, 8649))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8649.380");
| Modulus: | \(8649\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(8649\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(930\) |
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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);
|
\(\chi_{8649}(2,\cdot)\)
\(\chi_{8649}(47,\cdot)\)
\(\chi_{8649}(95,\cdot)\)
\(\chi_{8649}(101,\cdot)\)
\(\chi_{8649}(128,\cdot)\)
\(\chi_{8649}(140,\cdot)\)
\(\chi_{8649}(194,\cdot)\)
\(\chi_{8649}(221,\cdot)\)
\(\chi_{8649}(281,\cdot)\)
\(\chi_{8649}(326,\cdot)\)
\(\chi_{8649}(380,\cdot)\)
\(\chi_{8649}(407,\cdot)\)
\(\chi_{8649}(419,\cdot)\)
\(\chi_{8649}(473,\cdot)\)
\(\chi_{8649}(500,\cdot)\)
\(\chi_{8649}(560,\cdot)\)
\(\chi_{8649}(605,\cdot)\)
\(\chi_{8649}(653,\cdot)\)
\(\chi_{8649}(659,\cdot)\)
\(\chi_{8649}(686,\cdot)\)
\(\chi_{8649}(698,\cdot)\)
\(\chi_{8649}(752,\cdot)\)
\(\chi_{8649}(779,\cdot)\)
\(\chi_{8649}(839,\cdot)\)
\(\chi_{8649}(884,\cdot)\)
\(\chi_{8649}(932,\cdot)\)
\(\chi_{8649}(938,\cdot)\)
\(\chi_{8649}(965,\cdot)\)
\(\chi_{8649}(977,\cdot)\)
\(\chi_{8649}(1031,\cdot)\)
...
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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 };
\((3845,964)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{7}{155}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 8649 }(380, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{323}{930}\right)\) | \(e\left(\frac{323}{465}\right)\) | \(e\left(\frac{131}{186}\right)\) | \(e\left(\frac{178}{465}\right)\) | \(e\left(\frac{13}{310}\right)\) | \(e\left(\frac{8}{155}\right)\) | \(e\left(\frac{701}{930}\right)\) | \(e\left(\frac{206}{465}\right)\) | \(e\left(\frac{679}{930}\right)\) | \(e\left(\frac{181}{465}\right)\) |
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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)
