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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1256, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([78,78,7]))
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gp:[g,chi] = znchar(Mod(1195, 1256))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1256.1195");
| Modulus: | \(1256\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1256\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(156\) |
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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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\(\chi_{1256}(43,\cdot)\)
\(\chi_{1256}(83,\cdot)\)
\(\chi_{1256}(91,\cdot)\)
\(\chi_{1256}(123,\cdot)\)
\(\chi_{1256}(131,\cdot)\)
\(\chi_{1256}(139,\cdot)\)
\(\chi_{1256}(163,\cdot)\)
\(\chi_{1256}(195,\cdot)\)
\(\chi_{1256}(219,\cdot)\)
\(\chi_{1256}(227,\cdot)\)
\(\chi_{1256}(251,\cdot)\)
\(\chi_{1256}(259,\cdot)\)
\(\chi_{1256}(299,\cdot)\)
\(\chi_{1256}(387,\cdot)\)
\(\chi_{1256}(411,\cdot)\)
\(\chi_{1256}(451,\cdot)\)
\(\chi_{1256}(491,\cdot)\)
\(\chi_{1256}(531,\cdot)\)
\(\chi_{1256}(555,\cdot)\)
\(\chi_{1256}(643,\cdot)\)
\(\chi_{1256}(683,\cdot)\)
\(\chi_{1256}(691,\cdot)\)
\(\chi_{1256}(715,\cdot)\)
\(\chi_{1256}(723,\cdot)\)
\(\chi_{1256}(747,\cdot)\)
\(\chi_{1256}(779,\cdot)\)
\(\chi_{1256}(803,\cdot)\)
\(\chi_{1256}(811,\cdot)\)
\(\chi_{1256}(819,\cdot)\)
\(\chi_{1256}(851,\cdot)\)
...
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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 };
\((943,629,633)\) → \((-1,-1,e\left(\frac{7}{156}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1256 }(1195, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{85}{156}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{121}{156}\right)\) |
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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)
