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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12325, base_ring=CyclotomicField(560))
M = H._module
chi = DirichletCharacter(H, M([476,385,300]))
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gp:[g,chi] = znchar(Mod(1622, 12325))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12325.1622");
| Modulus: | \(12325\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12325\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(560\) |
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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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\(\chi_{12325}(3,\cdot)\)
\(\chi_{12325}(27,\cdot)\)
\(\chi_{12325}(48,\cdot)\)
\(\chi_{12325}(142,\cdot)\)
\(\chi_{12325}(148,\cdot)\)
\(\chi_{12325}(317,\cdot)\)
\(\chi_{12325}(388,\cdot)\)
\(\chi_{12325}(537,\cdot)\)
\(\chi_{12325}(583,\cdot)\)
\(\chi_{12325}(728,\cdot)\)
\(\chi_{12325}(772,\cdot)\)
\(\chi_{12325}(822,\cdot)\)
\(\chi_{12325}(827,\cdot)\)
\(\chi_{12325}(913,\cdot)\)
\(\chi_{12325}(1112,\cdot)\)
\(\chi_{12325}(1197,\cdot)\)
\(\chi_{12325}(1278,\cdot)\)
\(\chi_{12325}(1302,\cdot)\)
\(\chi_{12325}(1348,\cdot)\)
\(\chi_{12325}(1423,\cdot)\)
\(\chi_{12325}(1592,\cdot)\)
\(\chi_{12325}(1622,\cdot)\)
\(\chi_{12325}(1642,\cdot)\)
\(\chi_{12325}(1703,\cdot)\)
\(\chi_{12325}(1788,\cdot)\)
\(\chi_{12325}(1842,\cdot)\)
\(\chi_{12325}(1848,\cdot)\)
\(\chi_{12325}(1858,\cdot)\)
\(\chi_{12325}(1933,\cdot)\)
\(\chi_{12325}(2003,\cdot)\)
...
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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 };
\((3452,7976,11051)\) → \((e\left(\frac{17}{20}\right),e\left(\frac{11}{16}\right),e\left(\frac{15}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 12325 }(1622, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{3}{280}\right)\) | \(e\left(\frac{177}{560}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{183}{560}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{9}{280}\right)\) | \(e\left(\frac{177}{280}\right)\) | \(e\left(\frac{451}{560}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{19}{35}\right)\) |
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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)
