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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10309, base_ring=CyclotomicField(260))
M = H._module
chi = DirichletCharacter(H, M([160,247]))
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gp:[g,chi] = znchar(Mod(2341, 10309))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10309.2341");
| Modulus: | \(10309\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10309\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(260\) |
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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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\(\chi_{10309}(53,\cdot)\)
\(\chi_{10309}(313,\cdot)\)
\(\chi_{10309}(404,\cdot)\)
\(\chi_{10309}(521,\cdot)\)
\(\chi_{10309}(573,\cdot)\)
\(\chi_{10309}(586,\cdot)\)
\(\chi_{10309}(638,\cdot)\)
\(\chi_{10309}(755,\cdot)\)
\(\chi_{10309}(1106,\cdot)\)
\(\chi_{10309}(1197,\cdot)\)
\(\chi_{10309}(1314,\cdot)\)
\(\chi_{10309}(1366,\cdot)\)
\(\chi_{10309}(1379,\cdot)\)
\(\chi_{10309}(1431,\cdot)\)
\(\chi_{10309}(1548,\cdot)\)
\(\chi_{10309}(1639,\cdot)\)
\(\chi_{10309}(1899,\cdot)\)
\(\chi_{10309}(1990,\cdot)\)
\(\chi_{10309}(2107,\cdot)\)
\(\chi_{10309}(2159,\cdot)\)
\(\chi_{10309}(2172,\cdot)\)
\(\chi_{10309}(2224,\cdot)\)
\(\chi_{10309}(2341,\cdot)\)
\(\chi_{10309}(2432,\cdot)\)
\(\chi_{10309}(2692,\cdot)\)
\(\chi_{10309}(2783,\cdot)\)
\(\chi_{10309}(2900,\cdot)\)
\(\chi_{10309}(2952,\cdot)\)
\(\chi_{10309}(2965,\cdot)\)
\(\chi_{10309}(3017,\cdot)\)
...
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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 };
\((8114,2198)\) → \((e\left(\frac{8}{13}\right),e\left(\frac{19}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 10309 }(2341, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{147}{260}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{17}{130}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{149}{260}\right)\) | \(e\left(\frac{103}{260}\right)\) | \(e\left(\frac{181}{260}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{1}{260}\right)\) | \(e\left(\frac{33}{52}\right)\) |
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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)
