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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35090, base_ring=CyclotomicField(308))
M = H._module
chi = DirichletCharacter(H, M([231,252,143]))
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gp:[g,chi] = znchar(Mod(4973, 35090))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("35090.4973");
| Modulus: | \(35090\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(17545\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(308\) |
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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 \(\chi_{17545}(4973,\cdot)\) |
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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);
|
\(\chi_{35090}(287,\cdot)\)
\(\chi_{35090}(617,\cdot)\)
\(\chi_{35090}(1123,\cdot)\)
\(\chi_{35090}(1497,\cdot)\)
\(\chi_{35090}(1783,\cdot)\)
\(\chi_{35090}(2003,\cdot)\)
\(\chi_{35090}(2223,\cdot)\)
\(\chi_{35090}(2707,\cdot)\)
\(\chi_{35090}(2927,\cdot)\)
\(\chi_{35090}(3433,\cdot)\)
\(\chi_{35090}(3477,\cdot)\)
\(\chi_{35090}(3807,\cdot)\)
\(\chi_{35090}(4313,\cdot)\)
\(\chi_{35090}(4643,\cdot)\)
\(\chi_{35090}(4687,\cdot)\)
\(\chi_{35090}(4973,\cdot)\)
\(\chi_{35090}(5193,\cdot)\)
\(\chi_{35090}(5413,\cdot)\)
\(\chi_{35090}(5897,\cdot)\)
\(\chi_{35090}(6117,\cdot)\)
\(\chi_{35090}(6337,\cdot)\)
\(\chi_{35090}(6623,\cdot)\)
\(\chi_{35090}(6667,\cdot)\)
\(\chi_{35090}(6997,\cdot)\)
\(\chi_{35090}(7833,\cdot)\)
\(\chi_{35090}(7877,\cdot)\)
\(\chi_{35090}(8163,\cdot)\)
\(\chi_{35090}(8383,\cdot)\)
\(\chi_{35090}(8603,\cdot)\)
\(\chi_{35090}(9087,\cdot)\)
...
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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 };
\((14037,16821,21781)\) → \((-i,e\left(\frac{9}{11}\right),e\left(\frac{13}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(31\) |
| \( \chi_{ 35090 }(4973, a) \) |
\(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{15}{308}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{75}{308}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{181}{308}\right)\) | \(e\left(\frac{191}{308}\right)\) | \(e\left(\frac{249}{308}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{255}{308}\right)\) |
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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)
