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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(34645, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([585,535,234]))
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gp:[g,chi] = znchar(Mod(4063, 34645))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("34645.4063");
| Modulus: | \(34645\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(34645\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(780\) |
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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: | even |
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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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\(\chi_{34645}(228,\cdot)\)
\(\chi_{34645}(392,\cdot)\)
\(\chi_{34645}(843,\cdot)\)
\(\chi_{34645}(1007,\cdot)\)
\(\chi_{34645}(1138,\cdot)\)
\(\chi_{34645}(1302,\cdot)\)
\(\chi_{34645}(1398,\cdot)\)
\(\chi_{34645}(1562,\cdot)\)
\(\chi_{34645}(1753,\cdot)\)
\(\chi_{34645}(1788,\cdot)\)
\(\chi_{34645}(1917,\cdot)\)
\(\chi_{34645}(1952,\cdot)\)
\(\chi_{34645}(2013,\cdot)\)
\(\chi_{34645}(2177,\cdot)\)
\(\chi_{34645}(2403,\cdot)\)
\(\chi_{34645}(2567,\cdot)\)
\(\chi_{34645}(2893,\cdot)\)
\(\chi_{34645}(3057,\cdot)\)
\(\chi_{34645}(3508,\cdot)\)
\(\chi_{34645}(3672,\cdot)\)
\(\chi_{34645}(3803,\cdot)\)
\(\chi_{34645}(4063,\cdot)\)
\(\chi_{34645}(4227,\cdot)\)
\(\chi_{34645}(4418,\cdot)\)
\(\chi_{34645}(4453,\cdot)\)
\(\chi_{34645}(4617,\cdot)\)
\(\chi_{34645}(4678,\cdot)\)
\(\chi_{34645}(4842,\cdot)\)
\(\chi_{34645}(5068,\cdot)\)
\(\chi_{34645}(5232,\cdot)\)
...
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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 };
\((27717,22141,27886)\) → \((-i,e\left(\frac{107}{156}\right),e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 34645 }(4063, a) \) |
\(1\) | \(1\) | \(e\left(\frac{46}{195}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{92}{195}\right)\) | \(e\left(\frac{29}{780}\right)\) | \(e\left(\frac{164}{195}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{427}{780}\right)\) | \(e\left(\frac{71}{260}\right)\) | \(e\left(\frac{1}{13}\right)\) |
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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)
