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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3971, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([0,20]))
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gp:[g,chi] = znchar(Mod(2344, 3971))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3971.2344");
| Modulus: | \(3971\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(361\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(57\) |
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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 \(\chi_{361}(178,\cdot)\) |
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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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\(\chi_{3971}(45,\cdot)\)
\(\chi_{3971}(144,\cdot)\)
\(\chi_{3971}(254,\cdot)\)
\(\chi_{3971}(353,\cdot)\)
\(\chi_{3971}(463,\cdot)\)
\(\chi_{3971}(562,\cdot)\)
\(\chi_{3971}(672,\cdot)\)
\(\chi_{3971}(771,\cdot)\)
\(\chi_{3971}(881,\cdot)\)
\(\chi_{3971}(980,\cdot)\)
\(\chi_{3971}(1090,\cdot)\)
\(\chi_{3971}(1189,\cdot)\)
\(\chi_{3971}(1299,\cdot)\)
\(\chi_{3971}(1398,\cdot)\)
\(\chi_{3971}(1508,\cdot)\)
\(\chi_{3971}(1607,\cdot)\)
\(\chi_{3971}(1717,\cdot)\)
\(\chi_{3971}(1816,\cdot)\)
\(\chi_{3971}(1926,\cdot)\)
\(\chi_{3971}(2025,\cdot)\)
\(\chi_{3971}(2135,\cdot)\)
\(\chi_{3971}(2344,\cdot)\)
\(\chi_{3971}(2443,\cdot)\)
\(\chi_{3971}(2553,\cdot)\)
\(\chi_{3971}(2652,\cdot)\)
\(\chi_{3971}(2762,\cdot)\)
\(\chi_{3971}(2861,\cdot)\)
\(\chi_{3971}(2971,\cdot)\)
\(\chi_{3971}(3070,\cdot)\)
\(\chi_{3971}(3279,\cdot)\)
...
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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 };
\((1806,2168)\) → \((1,e\left(\frac{10}{57}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3971 }(2344, a) \) |
\(1\) | \(1\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) |
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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)
