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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(62465, base_ring=CyclotomicField(372))
M = H._module
chi = DirichletCharacter(H, M([279,310,34]))
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gp:[g,chi] = znchar(Mod(37268, 62465))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("62465.37268");
| Modulus: | \(62465\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(62465\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(372\) |
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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);
|
\(\chi_{62465}(192,\cdot)\)
\(\chi_{62465}(212,\cdot)\)
\(\chi_{62465}(998,\cdot)\)
\(\chi_{62465}(1018,\cdot)\)
\(\chi_{62465}(2207,\cdot)\)
\(\chi_{62465}(2227,\cdot)\)
\(\chi_{62465}(3013,\cdot)\)
\(\chi_{62465}(3033,\cdot)\)
\(\chi_{62465}(4222,\cdot)\)
\(\chi_{62465}(4242,\cdot)\)
\(\chi_{62465}(5028,\cdot)\)
\(\chi_{62465}(5048,\cdot)\)
\(\chi_{62465}(6237,\cdot)\)
\(\chi_{62465}(6257,\cdot)\)
\(\chi_{62465}(7043,\cdot)\)
\(\chi_{62465}(7063,\cdot)\)
\(\chi_{62465}(8252,\cdot)\)
\(\chi_{62465}(8272,\cdot)\)
\(\chi_{62465}(9058,\cdot)\)
\(\chi_{62465}(9078,\cdot)\)
\(\chi_{62465}(10267,\cdot)\)
\(\chi_{62465}(10287,\cdot)\)
\(\chi_{62465}(11073,\cdot)\)
\(\chi_{62465}(12282,\cdot)\)
\(\chi_{62465}(12302,\cdot)\)
\(\chi_{62465}(13088,\cdot)\)
\(\chi_{62465}(13108,\cdot)\)
\(\chi_{62465}(14297,\cdot)\)
\(\chi_{62465}(14317,\cdot)\)
\(\chi_{62465}(15103,\cdot)\)
...
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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 };
\((24987,24026,26911)\) → \((-i,e\left(\frac{5}{6}\right),e\left(\frac{17}{186}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 62465 }(37268, a) \) |
\(1\) | \(1\) | \(e\left(\frac{229}{372}\right)\) | \(e\left(\frac{251}{372}\right)\) | \(e\left(\frac{43}{186}\right)\) | \(e\left(\frac{9}{31}\right)\) | \(e\left(\frac{7}{124}\right)\) | \(e\left(\frac{105}{124}\right)\) | \(e\left(\frac{65}{186}\right)\) | \(e\left(\frac{11}{31}\right)\) | \(e\left(\frac{337}{372}\right)\) | \(e\left(\frac{125}{186}\right)\) |
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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)
