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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([92]))
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gp:[g,chi] = znchar(Mod(169, 242))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("242.169");
| Modulus: | \(242\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(121\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(55\) |
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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 \(\chi_{121}(48,\cdot)\) |
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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);
|
\(\chi_{242}(5,\cdot)\)
\(\chi_{242}(15,\cdot)\)
\(\chi_{242}(25,\cdot)\)
\(\chi_{242}(31,\cdot)\)
\(\chi_{242}(37,\cdot)\)
\(\chi_{242}(47,\cdot)\)
\(\chi_{242}(49,\cdot)\)
\(\chi_{242}(53,\cdot)\)
\(\chi_{242}(59,\cdot)\)
\(\chi_{242}(69,\cdot)\)
\(\chi_{242}(71,\cdot)\)
\(\chi_{242}(75,\cdot)\)
\(\chi_{242}(91,\cdot)\)
\(\chi_{242}(93,\cdot)\)
\(\chi_{242}(97,\cdot)\)
\(\chi_{242}(103,\cdot)\)
\(\chi_{242}(113,\cdot)\)
\(\chi_{242}(115,\cdot)\)
\(\chi_{242}(119,\cdot)\)
\(\chi_{242}(125,\cdot)\)
\(\chi_{242}(135,\cdot)\)
\(\chi_{242}(137,\cdot)\)
\(\chi_{242}(141,\cdot)\)
\(\chi_{242}(147,\cdot)\)
\(\chi_{242}(157,\cdot)\)
\(\chi_{242}(159,\cdot)\)
\(\chi_{242}(163,\cdot)\)
\(\chi_{242}(169,\cdot)\)
\(\chi_{242}(179,\cdot)\)
\(\chi_{242}(181,\cdot)\)
...
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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 };
\(123\) → \(e\left(\frac{46}{55}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 242 }(169, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
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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)
