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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2521, base_ring=CyclotomicField(280))
M = H._module
chi = DirichletCharacter(H, M([171]))
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gp:[g,chi] = znchar(Mod(233, 2521))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2521.233");
| Modulus: | \(2521\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2521\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(280\) |
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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: | odd |
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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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\(\chi_{2521}(29,\cdot)\)
\(\chi_{2521}(37,\cdot)\)
\(\chi_{2521}(47,\cdot)\)
\(\chi_{2521}(67,\cdot)\)
\(\chi_{2521}(89,\cdot)\)
\(\chi_{2521}(106,\cdot)\)
\(\chi_{2521}(221,\cdot)\)
\(\chi_{2521}(233,\cdot)\)
\(\chi_{2521}(278,\cdot)\)
\(\chi_{2521}(285,\cdot)\)
\(\chi_{2521}(298,\cdot)\)
\(\chi_{2521}(299,\cdot)\)
\(\chi_{2521}(301,\cdot)\)
\(\chi_{2521}(308,\cdot)\)
\(\chi_{2521}(331,\cdot)\)
\(\chi_{2521}(430,\cdot)\)
\(\chi_{2521}(440,\cdot)\)
\(\chi_{2521}(462,\cdot)\)
\(\chi_{2521}(547,\cdot)\)
\(\chi_{2521}(565,\cdot)\)
\(\chi_{2521}(590,\cdot)\)
\(\chi_{2521}(655,\cdot)\)
\(\chi_{2521}(660,\cdot)\)
\(\chi_{2521}(685,\cdot)\)
\(\chi_{2521}(693,\cdot)\)
\(\chi_{2521}(736,\cdot)\)
\(\chi_{2521}(764,\cdot)\)
\(\chi_{2521}(802,\cdot)\)
\(\chi_{2521}(812,\cdot)\)
\(\chi_{2521}(816,\cdot)\)
...
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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 };
\(17\) → \(e\left(\frac{171}{280}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 2521 }(233, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{9}{56}\right)\) |
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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)
