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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4761, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([46,45]))
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gp:[g,chi] = znchar(Mod(229, 4761))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4761.229");
| Modulus: | \(4761\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(4761\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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);
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\(\chi_{4761}(22,\cdot)\)
\(\chi_{4761}(160,\cdot)\)
\(\chi_{4761}(229,\cdot)\)
\(\chi_{4761}(367,\cdot)\)
\(\chi_{4761}(436,\cdot)\)
\(\chi_{4761}(574,\cdot)\)
\(\chi_{4761}(643,\cdot)\)
\(\chi_{4761}(781,\cdot)\)
\(\chi_{4761}(850,\cdot)\)
\(\chi_{4761}(988,\cdot)\)
\(\chi_{4761}(1195,\cdot)\)
\(\chi_{4761}(1264,\cdot)\)
\(\chi_{4761}(1402,\cdot)\)
\(\chi_{4761}(1471,\cdot)\)
\(\chi_{4761}(1609,\cdot)\)
\(\chi_{4761}(1678,\cdot)\)
\(\chi_{4761}(1816,\cdot)\)
\(\chi_{4761}(1885,\cdot)\)
\(\chi_{4761}(2023,\cdot)\)
\(\chi_{4761}(2092,\cdot)\)
\(\chi_{4761}(2230,\cdot)\)
\(\chi_{4761}(2299,\cdot)\)
\(\chi_{4761}(2437,\cdot)\)
\(\chi_{4761}(2506,\cdot)\)
\(\chi_{4761}(2713,\cdot)\)
\(\chi_{4761}(2851,\cdot)\)
\(\chi_{4761}(2920,\cdot)\)
\(\chi_{4761}(3058,\cdot)\)
\(\chi_{4761}(3127,\cdot)\)
\(\chi_{4761}(3265,\cdot)\)
...
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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 };
\((2117,1063)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{15}{46}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 4761 }(229, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{137}{138}\right)\) | \(e\left(\frac{55}{138}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{103}{138}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{131}{138}\right)\) | \(e\left(\frac{14}{69}\right)\) |
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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)
