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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10309, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([450,611]))
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gp:[g,chi] = znchar(Mod(1481, 10309))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10309.1481");
| 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: | \(780\) |
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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);
|
\(\chi_{10309}(51,\cdot)\)
\(\chi_{10309}(116,\cdot)\)
\(\chi_{10309}(129,\cdot)\)
\(\chi_{10309}(181,\cdot)\)
\(\chi_{10309}(246,\cdot)\)
\(\chi_{10309}(298,\cdot)\)
\(\chi_{10309}(311,\cdot)\)
\(\chi_{10309}(376,\cdot)\)
\(\chi_{10309}(519,\cdot)\)
\(\chi_{10309}(532,\cdot)\)
\(\chi_{10309}(584,\cdot)\)
\(\chi_{10309}(636,\cdot)\)
\(\chi_{10309}(688,\cdot)\)
\(\chi_{10309}(701,\cdot)\)
\(\chi_{10309}(714,\cdot)\)
\(\chi_{10309}(909,\cdot)\)
\(\chi_{10309}(922,\cdot)\)
\(\chi_{10309}(974,\cdot)\)
\(\chi_{10309}(1039,\cdot)\)
\(\chi_{10309}(1091,\cdot)\)
\(\chi_{10309}(1104,\cdot)\)
\(\chi_{10309}(1169,\cdot)\)
\(\chi_{10309}(1299,\cdot)\)
\(\chi_{10309}(1312,\cdot)\)
\(\chi_{10309}(1325,\cdot)\)
\(\chi_{10309}(1377,\cdot)\)
\(\chi_{10309}(1429,\cdot)\)
\(\chi_{10309}(1481,\cdot)\)
\(\chi_{10309}(1494,\cdot)\)
\(\chi_{10309}(1507,\cdot)\)
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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 };
\((8114,2198)\) → \((e\left(\frac{15}{26}\right),e\left(\frac{47}{60}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 10309 }(1481, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{281}{780}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{281}{390}\right)\) | \(e\left(\frac{83}{195}\right)\) | \(e\left(\frac{467}{780}\right)\) | \(e\left(\frac{89}{780}\right)\) | \(e\left(\frac{21}{260}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{613}{780}\right)\) | \(e\left(\frac{9}{52}\right)\) |
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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)
