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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61225, base_ring=CyclotomicField(130))
M = H._module
chi = DirichletCharacter(H, M([65,39,95]))
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gp:[g,chi] = znchar(Mod(13049, 61225))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61225.13049");
| Modulus: | \(61225\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12245\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(130\) |
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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 \(\chi_{12245}(804,\cdot)\) |
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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);
|
\(\chi_{61225}(649,\cdot)\)
\(\chi_{61225}(1449,\cdot)\)
\(\chi_{61225}(2224,\cdot)\)
\(\chi_{61225}(3774,\cdot)\)
\(\chi_{61225}(4324,\cdot)\)
\(\chi_{61225}(5824,\cdot)\)
\(\chi_{61225}(7374,\cdot)\)
\(\chi_{61225}(8149,\cdot)\)
\(\chi_{61225}(9699,\cdot)\)
\(\chi_{61225}(10524,\cdot)\)
\(\chi_{61225}(11524,\cdot)\)
\(\chi_{61225}(13049,\cdot)\)
\(\chi_{61225}(16149,\cdot)\)
\(\chi_{61225}(16174,\cdot)\)
\(\chi_{61225}(17449,\cdot)\)
\(\chi_{61225}(22099,\cdot)\)
\(\chi_{61225}(22349,\cdot)\)
\(\chi_{61225}(22374,\cdot)\)
\(\chi_{61225}(22924,\cdot)\)
\(\chi_{61225}(23899,\cdot)\)
\(\chi_{61225}(26024,\cdot)\)
\(\chi_{61225}(28299,\cdot)\)
\(\chi_{61225}(29324,\cdot)\)
\(\chi_{61225}(32224,\cdot)\)
\(\chi_{61225}(33774,\cdot)\)
\(\chi_{61225}(34774,\cdot)\)
\(\chi_{61225}(37874,\cdot)\)
\(\chi_{61225}(39199,\cdot)\)
\(\chi_{61225}(39399,\cdot)\)
\(\chi_{61225}(40699,\cdot)\)
...
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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 };
\((58777,33576,32551)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{19}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 61225 }(13049, a) \) |
\(1\) | \(1\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{42}{65}\right)\) |
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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)
