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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,56,15]))
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gp:[g,chi] = znchar(Mod(1217, 1309))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1309.1217");
| Modulus: | \(1309\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1309\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(80\) |
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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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\(\chi_{1309}(6,\cdot)\)
\(\chi_{1309}(41,\cdot)\)
\(\chi_{1309}(62,\cdot)\)
\(\chi_{1309}(90,\cdot)\)
\(\chi_{1309}(139,\cdot)\)
\(\chi_{1309}(160,\cdot)\)
\(\chi_{1309}(167,\cdot)\)
\(\chi_{1309}(216,\cdot)\)
\(\chi_{1309}(244,\cdot)\)
\(\chi_{1309}(398,\cdot)\)
\(\chi_{1309}(447,\cdot)\)
\(\chi_{1309}(503,\cdot)\)
\(\chi_{1309}(524,\cdot)\)
\(\chi_{1309}(601,\cdot)\)
\(\chi_{1309}(622,\cdot)\)
\(\chi_{1309}(657,\cdot)\)
\(\chi_{1309}(734,\cdot)\)
\(\chi_{1309}(755,\cdot)\)
\(\chi_{1309}(776,\cdot)\)
\(\chi_{1309}(811,\cdot)\)
\(\chi_{1309}(853,\cdot)\)
\(\chi_{1309}(860,\cdot)\)
\(\chi_{1309}(930,\cdot)\)
\(\chi_{1309}(1014,\cdot)\)
\(\chi_{1309}(1042,\cdot)\)
\(\chi_{1309}(1091,\cdot)\)
\(\chi_{1309}(1119,\cdot)\)
\(\chi_{1309}(1161,\cdot)\)
\(\chi_{1309}(1168,\cdot)\)
\(\chi_{1309}(1196,\cdot)\)
...
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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 };
\((1123,596,309)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{3}{16}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 1309 }(1217, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{19}{20}\right)\) |
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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)
