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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(28611, base_ring=CyclotomicField(4080))
M = H._module
chi = DirichletCharacter(H, M([3400,2856,1635]))
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gp:[g,chi] = znchar(Mod(590, 28611))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("28611.590");
| Modulus: | \(28611\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(28611\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(4080\) |
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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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\(\chi_{28611}(29,\cdot)\)
\(\chi_{28611}(41,\cdot)\)
\(\chi_{28611}(74,\cdot)\)
\(\chi_{28611}(95,\cdot)\)
\(\chi_{28611}(167,\cdot)\)
\(\chi_{28611}(173,\cdot)\)
\(\chi_{28611}(182,\cdot)\)
\(\chi_{28611}(194,\cdot)\)
\(\chi_{28611}(227,\cdot)\)
\(\chi_{28611}(248,\cdot)\)
\(\chi_{28611}(266,\cdot)\)
\(\chi_{28611}(299,\cdot)\)
\(\chi_{28611}(326,\cdot)\)
\(\chi_{28611}(347,\cdot)\)
\(\chi_{28611}(371,\cdot)\)
\(\chi_{28611}(380,\cdot)\)
\(\chi_{28611}(398,\cdot)\)
\(\chi_{28611}(437,\cdot)\)
\(\chi_{28611}(464,\cdot)\)
\(\chi_{28611}(470,\cdot)\)
\(\chi_{28611}(479,\cdot)\)
\(\chi_{28611}(524,\cdot)\)
\(\chi_{28611}(590,\cdot)\)
\(\chi_{28611}(623,\cdot)\)
\(\chi_{28611}(635,\cdot)\)
\(\chi_{28611}(668,\cdot)\)
\(\chi_{28611}(677,\cdot)\)
\(\chi_{28611}(734,\cdot)\)
\(\chi_{28611}(743,\cdot)\)
\(\chi_{28611}(776,\cdot)\)
...
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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 };
\((15896,23410,7228)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{10}\right),e\left(\frac{109}{272}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 28611 }(590, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1373}{2040}\right)\) | \(e\left(\frac{353}{1020}\right)\) | \(e\left(\frac{2999}{4080}\right)\) | \(e\left(\frac{1417}{4080}\right)\) | \(e\left(\frac{13}{680}\right)\) | \(e\left(\frac{111}{272}\right)\) | \(e\left(\frac{929}{1020}\right)\) | \(e\left(\frac{83}{4080}\right)\) | \(e\left(\frac{353}{510}\right)\) | \(e\left(\frac{483}{680}\right)\) |
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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)
