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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20433, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([69,69,49]))
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gp:[g,chi] = znchar(Mod(6467, 20433))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("20433.6467");
| Modulus: | \(20433\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2919\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(138\) |
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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 \(\chi_{2919}(629,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{20433}(293,\cdot)\)
\(\chi_{20433}(2057,\cdot)\)
\(\chi_{20433}(2204,\cdot)\)
\(\chi_{20433}(2498,\cdot)\)
\(\chi_{20433}(2792,\cdot)\)
\(\chi_{20433}(3821,\cdot)\)
\(\chi_{20433}(4262,\cdot)\)
\(\chi_{20433}(4556,\cdot)\)
\(\chi_{20433}(4997,\cdot)\)
\(\chi_{20433}(5438,\cdot)\)
\(\chi_{20433}(6467,\cdot)\)
\(\chi_{20433}(8084,\cdot)\)
\(\chi_{20433}(8966,\cdot)\)
\(\chi_{20433}(9554,\cdot)\)
\(\chi_{20433}(9701,\cdot)\)
\(\chi_{20433}(9995,\cdot)\)
\(\chi_{20433}(10142,\cdot)\)
\(\chi_{20433}(10289,\cdot)\)
\(\chi_{20433}(10583,\cdot)\)
\(\chi_{20433}(12347,\cdot)\)
\(\chi_{20433}(12494,\cdot)\)
\(\chi_{20433}(13376,\cdot)\)
\(\chi_{20433}(13523,\cdot)\)
\(\chi_{20433}(13817,\cdot)\)
\(\chi_{20433}(14111,\cdot)\)
\(\chi_{20433}(14405,\cdot)\)
\(\chi_{20433}(14699,\cdot)\)
\(\chi_{20433}(15140,\cdot)\)
\(\chi_{20433}(15728,\cdot)\)
\(\chi_{20433}(16316,\cdot)\)
...
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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 };
\((6812,1669,12790)\) → \((-1,-1,e\left(\frac{49}{138}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 20433 }(6467, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{67}{138}\right)\) | \(e\left(\frac{31}{138}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{137}{138}\right)\) | \(e\left(\frac{11}{69}\right)\) |
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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)
