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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39195, base_ring=CyclotomicField(132))
M = H._module
chi = DirichletCharacter(H, M([66,99,121,10]))
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gp:[g,chi] = znchar(Mod(15308, 39195))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39195.15308");
| Modulus: | \(39195\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13065\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(132\) |
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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 \(\chi_{13065}(2243,\cdot)\) |
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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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\(\chi_{39195}(98,\cdot)\)
\(\chi_{39195}(422,\cdot)\)
\(\chi_{39195}(2663,\cdot)\)
\(\chi_{39195}(3932,\cdot)\)
\(\chi_{39195}(6272,\cdot)\)
\(\chi_{39195}(6533,\cdot)\)
\(\chi_{39195}(7217,\cdot)\)
\(\chi_{39195}(8972,\cdot)\)
\(\chi_{39195}(9458,\cdot)\)
\(\chi_{39195}(10952,\cdot)\)
\(\chi_{39195}(11438,\cdot)\)
\(\chi_{39195}(13067,\cdot)\)
\(\chi_{39195}(13193,\cdot)\)
\(\chi_{39195}(13652,\cdot)\)
\(\chi_{39195}(15308,\cdot)\)
\(\chi_{39195}(16478,\cdot)\)
\(\chi_{39195}(16577,\cdot)\)
\(\chi_{39195}(17288,\cdot)\)
\(\chi_{39195}(17387,\cdot)\)
\(\chi_{39195}(17873,\cdot)\)
\(\chi_{39195}(20312,\cdot)\)
\(\chi_{39195}(20798,\cdot)\)
\(\chi_{39195}(21257,\cdot)\)
\(\chi_{39195}(23597,\cdot)\)
\(\chi_{39195}(24668,\cdot)\)
\(\chi_{39195}(24767,\cdot)\)
\(\chi_{39195}(25478,\cdot)\)
\(\chi_{39195}(26162,\cdot)\)
\(\chi_{39195}(26522,\cdot)\)
\(\chi_{39195}(27008,\cdot)\)
...
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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 };
\((34841,31357,36181,15211)\) → \((-1,-i,e\left(\frac{11}{12}\right),e\left(\frac{5}{66}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 39195 }(15308, a) \) |
\(1\) | \(1\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{83}{132}\right)\) |
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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)
