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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2848, base_ring=CyclotomicField(88))
M = H._module
chi = DirichletCharacter(H, M([0,11,80]))
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gp:[g,chi] = znchar(Mod(1189, 2848))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2848.1189");
| Modulus: | \(2848\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2848\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(88\) |
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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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\(\chi_{2848}(45,\cdot)\)
\(\chi_{2848}(93,\cdot)\)
\(\chi_{2848}(245,\cdot)\)
\(\chi_{2848}(269,\cdot)\)
\(\chi_{2848}(453,\cdot)\)
\(\chi_{2848}(461,\cdot)\)
\(\chi_{2848}(477,\cdot)\)
\(\chi_{2848}(509,\cdot)\)
\(\chi_{2848}(573,\cdot)\)
\(\chi_{2848}(701,\cdot)\)
\(\chi_{2848}(757,\cdot)\)
\(\chi_{2848}(805,\cdot)\)
\(\chi_{2848}(957,\cdot)\)
\(\chi_{2848}(981,\cdot)\)
\(\chi_{2848}(1165,\cdot)\)
\(\chi_{2848}(1173,\cdot)\)
\(\chi_{2848}(1189,\cdot)\)
\(\chi_{2848}(1221,\cdot)\)
\(\chi_{2848}(1285,\cdot)\)
\(\chi_{2848}(1413,\cdot)\)
\(\chi_{2848}(1469,\cdot)\)
\(\chi_{2848}(1517,\cdot)\)
\(\chi_{2848}(1669,\cdot)\)
\(\chi_{2848}(1693,\cdot)\)
\(\chi_{2848}(1877,\cdot)\)
\(\chi_{2848}(1885,\cdot)\)
\(\chi_{2848}(1901,\cdot)\)
\(\chi_{2848}(1933,\cdot)\)
\(\chi_{2848}(1997,\cdot)\)
\(\chi_{2848}(2125,\cdot)\)
...
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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 };
\((1247,357,1249)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{10}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2848 }(1189, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{15}{88}\right)\) |
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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)
