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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([170,153]))
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gp:[g,chi] = znchar(Mod(122, 675))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("675.122");
| Modulus: | \(675\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(675\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(180\) |
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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);
|
\(\chi_{675}(2,\cdot)\)
\(\chi_{675}(23,\cdot)\)
\(\chi_{675}(38,\cdot)\)
\(\chi_{675}(47,\cdot)\)
\(\chi_{675}(77,\cdot)\)
\(\chi_{675}(83,\cdot)\)
\(\chi_{675}(92,\cdot)\)
\(\chi_{675}(113,\cdot)\)
\(\chi_{675}(122,\cdot)\)
\(\chi_{675}(128,\cdot)\)
\(\chi_{675}(137,\cdot)\)
\(\chi_{675}(158,\cdot)\)
\(\chi_{675}(167,\cdot)\)
\(\chi_{675}(173,\cdot)\)
\(\chi_{675}(203,\cdot)\)
\(\chi_{675}(212,\cdot)\)
\(\chi_{675}(227,\cdot)\)
\(\chi_{675}(248,\cdot)\)
\(\chi_{675}(263,\cdot)\)
\(\chi_{675}(272,\cdot)\)
\(\chi_{675}(302,\cdot)\)
\(\chi_{675}(308,\cdot)\)
\(\chi_{675}(317,\cdot)\)
\(\chi_{675}(338,\cdot)\)
\(\chi_{675}(347,\cdot)\)
\(\chi_{675}(353,\cdot)\)
\(\chi_{675}(362,\cdot)\)
\(\chi_{675}(383,\cdot)\)
\(\chi_{675}(392,\cdot)\)
\(\chi_{675}(398,\cdot)\)
...
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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 };
\((326,352)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{17}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 675 }(122, a) \) |
\(1\) | \(1\) | \(e\left(\frac{143}{180}\right)\) | \(e\left(\frac{53}{90}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{79}{90}\right)\) | \(e\left(\frac{127}{180}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{8}{45}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) |
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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)
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sage:chi.gauss_sum(a)
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gp:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
