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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74725, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([70,90,91]))
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gp:[g,chi] = znchar(Mod(28524, 74725))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74725.28524");
| Modulus: | \(74725\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(14945\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(140\) |
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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 \(\chi_{14945}(13579,\cdot)\) |
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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);
|
\(\chi_{74725}(699,\cdot)\)
\(\chi_{74725}(1924,\cdot)\)
\(\chi_{74725}(3149,\cdot)\)
\(\chi_{74725}(6474,\cdot)\)
\(\chi_{74725}(6824,\cdot)\)
\(\chi_{74725}(7174,\cdot)\)
\(\chi_{74725}(10149,\cdot)\)
\(\chi_{74725}(11374,\cdot)\)
\(\chi_{74725}(12599,\cdot)\)
\(\chi_{74725}(13824,\cdot)\)
\(\chi_{74725}(16799,\cdot)\)
\(\chi_{74725}(17499,\cdot)\)
\(\chi_{74725}(17849,\cdot)\)
\(\chi_{74725}(27474,\cdot)\)
\(\chi_{74725}(27824,\cdot)\)
\(\chi_{74725}(28524,\cdot)\)
\(\chi_{74725}(31499,\cdot)\)
\(\chi_{74725}(32724,\cdot)\)
\(\chi_{74725}(33949,\cdot)\)
\(\chi_{74725}(35174,\cdot)\)
\(\chi_{74725}(38149,\cdot)\)
\(\chi_{74725}(38499,\cdot)\)
\(\chi_{74725}(38849,\cdot)\)
\(\chi_{74725}(42174,\cdot)\)
\(\chi_{74725}(43399,\cdot)\)
\(\chi_{74725}(44624,\cdot)\)
\(\chi_{74725}(45849,\cdot)\)
\(\chi_{74725}(48824,\cdot)\)
\(\chi_{74725}(49174,\cdot)\)
\(\chi_{74725}(49524,\cdot)\)
...
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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 };
\((26902,50326,60026)\) → \((-1,e\left(\frac{9}{14}\right),e\left(\frac{13}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 74725 }(28524, a) \) |
\(1\) | \(1\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{16}{35}\right)\) |
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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)
