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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(39690, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([119,0,87]))
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gp:[g,chi] = znchar(Mod(4121, 39690))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("39690.4121");
| Modulus: | \(39690\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1323\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(126\) |
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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 \(\chi_{1323}(446,\cdot)\) |
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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: | no |
| 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);
|
\(\chi_{39690}(341,\cdot)\)
\(\chi_{39690}(2231,\cdot)\)
\(\chi_{39690}(2411,\cdot)\)
\(\chi_{39690}(4121,\cdot)\)
\(\chi_{39690}(4301,\cdot)\)
\(\chi_{39690}(6011,\cdot)\)
\(\chi_{39690}(6191,\cdot)\)
\(\chi_{39690}(7901,\cdot)\)
\(\chi_{39690}(8081,\cdot)\)
\(\chi_{39690}(9791,\cdot)\)
\(\chi_{39690}(9971,\cdot)\)
\(\chi_{39690}(11861,\cdot)\)
\(\chi_{39690}(13571,\cdot)\)
\(\chi_{39690}(15461,\cdot)\)
\(\chi_{39690}(15641,\cdot)\)
\(\chi_{39690}(17351,\cdot)\)
\(\chi_{39690}(17531,\cdot)\)
\(\chi_{39690}(19241,\cdot)\)
\(\chi_{39690}(19421,\cdot)\)
\(\chi_{39690}(21131,\cdot)\)
\(\chi_{39690}(21311,\cdot)\)
\(\chi_{39690}(23021,\cdot)\)
\(\chi_{39690}(23201,\cdot)\)
\(\chi_{39690}(25091,\cdot)\)
\(\chi_{39690}(26801,\cdot)\)
\(\chi_{39690}(28691,\cdot)\)
\(\chi_{39690}(28871,\cdot)\)
\(\chi_{39690}(30581,\cdot)\)
\(\chi_{39690}(30761,\cdot)\)
\(\chi_{39690}(32471,\cdot)\)
...
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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 };
\((29891,15877,27541)\) → \((e\left(\frac{17}{18}\right),1,e\left(\frac{29}{42}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 39690 }(4121, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(-1\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) |
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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)
