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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1687, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([0,113]))
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gp:[g,chi] = znchar(Mod(50, 1687))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1687.50");
| Modulus: | \(1687\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(241\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(120\) |
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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: | no, induced from \(\chi_{241}(50,\cdot)\) |
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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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\(\chi_{1687}(29,\cdot)\)
\(\chi_{1687}(50,\cdot)\)
\(\chi_{1687}(169,\cdot)\)
\(\chi_{1687}(253,\cdot)\)
\(\chi_{1687}(316,\cdot)\)
\(\chi_{1687}(407,\cdot)\)
\(\chi_{1687}(470,\cdot)\)
\(\chi_{1687}(554,\cdot)\)
\(\chi_{1687}(673,\cdot)\)
\(\chi_{1687}(694,\cdot)\)
\(\chi_{1687}(743,\cdot)\)
\(\chi_{1687}(897,\cdot)\)
\(\chi_{1687}(911,\cdot)\)
\(\chi_{1687}(946,\cdot)\)
\(\chi_{1687}(967,\cdot)\)
\(\chi_{1687}(1009,\cdot)\)
\(\chi_{1687}(1023,\cdot)\)
\(\chi_{1687}(1044,\cdot)\)
\(\chi_{1687}(1072,\cdot)\)
\(\chi_{1687}(1128,\cdot)\)
\(\chi_{1687}(1156,\cdot)\)
\(\chi_{1687}(1254,\cdot)\)
\(\chi_{1687}(1282,\cdot)\)
\(\chi_{1687}(1338,\cdot)\)
\(\chi_{1687}(1366,\cdot)\)
\(\chi_{1687}(1387,\cdot)\)
\(\chi_{1687}(1401,\cdot)\)
\(\chi_{1687}(1443,\cdot)\)
\(\chi_{1687}(1464,\cdot)\)
\(\chi_{1687}(1499,\cdot)\)
...
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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 };
\((724,1212)\) → \((1,e\left(\frac{113}{120}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1687 }(50, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{13}{60}\right)\) |
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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)
