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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80010, base_ring=CyclotomicField(252))
M = H._module
chi = DirichletCharacter(H, M([126,189,42,148]))
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gp:[g,chi] = znchar(Mod(15893, 80010))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80010.15893");
| Modulus: | \(80010\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(13335\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(252\) |
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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 \(\chi_{13335}(2558,\cdot)\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{80010}(17,\cdot)\)
\(\chi_{80010}(773,\cdot)\)
\(\chi_{80010}(1097,\cdot)\)
\(\chi_{80010}(2357,\cdot)\)
\(\chi_{80010}(2483,\cdot)\)
\(\chi_{80010}(3743,\cdot)\)
\(\chi_{80010}(5813,\cdot)\)
\(\chi_{80010}(6893,\cdot)\)
\(\chi_{80010}(7073,\cdot)\)
\(\chi_{80010}(7397,\cdot)\)
\(\chi_{80010}(7523,\cdot)\)
\(\chi_{80010}(7577,\cdot)\)
\(\chi_{80010}(8027,\cdot)\)
\(\chi_{80010}(8837,\cdot)\)
\(\chi_{80010}(9413,\cdot)\)
\(\chi_{80010}(9467,\cdot)\)
\(\chi_{80010}(11987,\cdot)\)
\(\chi_{80010}(12617,\cdot)\)
\(\chi_{80010}(13067,\cdot)\)
\(\chi_{80010}(13877,\cdot)\)
\(\chi_{80010}(15893,\cdot)\)
\(\chi_{80010}(17027,\cdot)\)
\(\chi_{80010}(19493,\cdot)\)
\(\chi_{80010}(25667,\cdot)\)
\(\chi_{80010}(27683,\cdot)\)
\(\chi_{80010}(28187,\cdot)\)
\(\chi_{80010}(28817,\cdot)\)
\(\chi_{80010}(28997,\cdot)\)
\(\chi_{80010}(29627,\cdot)\)
\(\chi_{80010}(30203,\cdot)\)
...
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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 };
\((35561,48007,57151,15751)\) → \((-1,-i,e\left(\frac{1}{6}\right),e\left(\frac{37}{63}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 80010 }(15893, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{241}{252}\right)\) | \(e\left(\frac{185}{252}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{252}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{239}{252}\right)\) |
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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)
