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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987696, base_ring=CyclotomicField(1444))
M = H._module
chi = DirichletCharacter(H, M([0,1083,0,1138]))
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gp:[g,chi] = znchar(Mod(42445, 987696))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("987696.42445");
| Modulus: | \(987696\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(109744\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1444\) |
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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 \(\chi_{109744}(42445,\cdot)\) |
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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);
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\(\chi_{987696}(37,\cdot)\)
\(\chi_{987696}(1405,\cdot)\)
\(\chi_{987696}(2773,\cdot)\)
\(\chi_{987696}(4141,\cdot)\)
\(\chi_{987696}(5509,\cdot)\)
\(\chi_{987696}(6877,\cdot)\)
\(\chi_{987696}(8245,\cdot)\)
\(\chi_{987696}(9613,\cdot)\)
\(\chi_{987696}(10981,\cdot)\)
\(\chi_{987696}(12349,\cdot)\)
\(\chi_{987696}(15085,\cdot)\)
\(\chi_{987696}(16453,\cdot)\)
\(\chi_{987696}(17821,\cdot)\)
\(\chi_{987696}(19189,\cdot)\)
\(\chi_{987696}(20557,\cdot)\)
\(\chi_{987696}(21925,\cdot)\)
\(\chi_{987696}(23293,\cdot)\)
\(\chi_{987696}(24661,\cdot)\)
\(\chi_{987696}(26029,\cdot)\)
\(\chi_{987696}(27397,\cdot)\)
\(\chi_{987696}(28765,\cdot)\)
\(\chi_{987696}(30133,\cdot)\)
\(\chi_{987696}(31501,\cdot)\)
\(\chi_{987696}(32869,\cdot)\)
\(\chi_{987696}(34237,\cdot)\)
\(\chi_{987696}(35605,\cdot)\)
\(\chi_{987696}(36973,\cdot)\)
\(\chi_{987696}(38341,\cdot)\)
\(\chi_{987696}(41077,\cdot)\)
\(\chi_{987696}(42445,\cdot)\)
...
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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 };
\((617311,740773,438977,857377)\) → \((1,-i,1,e\left(\frac{569}{722}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 987696 }(42445, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{1379}{1444}\right)\) | \(e\left(\frac{173}{722}\right)\) | \(e\left(\frac{119}{1444}\right)\) | \(e\left(\frac{615}{1444}\right)\) | \(e\left(\frac{210}{361}\right)\) | \(e\left(\frac{253}{722}\right)\) | \(e\left(\frac{657}{722}\right)\) | \(e\left(\frac{403}{1444}\right)\) | \(e\left(\frac{77}{722}\right)\) | \(e\left(\frac{281}{1444}\right)\) |
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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)
