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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2977, base_ring=CyclotomicField(228))
M = H._module
chi = DirichletCharacter(H, M([95,85]))
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gp:[g,chi] = znchar(Mod(1241, 2977))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2977.1241");
| Modulus: | \(2977\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2977\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(228\) |
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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: | even |
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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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\(\chi_{2977}(6,\cdot)\)
\(\chi_{2977}(28,\cdot)\)
\(\chi_{2977}(98,\cdot)\)
\(\chi_{2977}(162,\cdot)\)
\(\chi_{2977}(279,\cdot)\)
\(\chi_{2977}(288,\cdot)\)
\(\chi_{2977}(319,\cdot)\)
\(\chi_{2977}(331,\cdot)\)
\(\chi_{2977}(345,\cdot)\)
\(\chi_{2977}(379,\cdot)\)
\(\chi_{2977}(384,\cdot)\)
\(\chi_{2977}(423,\cdot)\)
\(\chi_{2977}(427,\cdot)\)
\(\chi_{2977}(487,\cdot)\)
\(\chi_{2977}(496,\cdot)\)
\(\chi_{2977}(505,\cdot)\)
\(\chi_{2977}(535,\cdot)\)
\(\chi_{2977}(570,\cdot)\)
\(\chi_{2977}(752,\cdot)\)
\(\chi_{2977}(756,\cdot)\)
\(\chi_{2977}(760,\cdot)\)
\(\chi_{2977}(955,\cdot)\)
\(\chi_{2977}(982,\cdot)\)
\(\chi_{2977}(1003,\cdot)\)
\(\chi_{2977}(1008,\cdot)\)
\(\chi_{2977}(1155,\cdot)\)
\(\chi_{2977}(1185,\cdot)\)
\(\chi_{2977}(1241,\cdot)\)
\(\chi_{2977}(1250,\cdot)\)
\(\chi_{2977}(1302,\cdot)\)
...
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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 };
\((1146,235)\) → \((e\left(\frac{5}{12}\right),e\left(\frac{85}{228}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2977 }(1241, a) \) |
\(1\) | \(1\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{65}{228}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{121}{228}\right)\) | \(e\left(\frac{71}{228}\right)\) |
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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)
