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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(18975, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,77,99,25]))
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gp:[g,chi] = znchar(Mod(2734, 18975))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("18975.2734");
| Modulus: | \(18975\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(6325\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(110\) |
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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 \(\chi_{6325}(2734,\cdot)\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{18975}(19,\cdot)\)
\(\chi_{18975}(304,\cdot)\)
\(\chi_{18975}(2389,\cdot)\)
\(\chi_{18975}(2494,\cdot)\)
\(\chi_{18975}(2734,\cdot)\)
\(\chi_{18975}(2779,\cdot)\)
\(\chi_{18975}(3214,\cdot)\)
\(\chi_{18975}(3319,\cdot)\)
\(\chi_{18975}(3559,\cdot)\)
\(\chi_{18975}(4039,\cdot)\)
\(\chi_{18975}(4384,\cdot)\)
\(\chi_{18975}(4864,\cdot)\)
\(\chi_{18975}(5209,\cdot)\)
\(\chi_{18975}(5254,\cdot)\)
\(\chi_{18975}(5794,\cdot)\)
\(\chi_{18975}(6079,\cdot)\)
\(\chi_{18975}(6514,\cdot)\)
\(\chi_{18975}(6859,\cdot)\)
\(\chi_{18975}(7444,\cdot)\)
\(\chi_{18975}(8554,\cdot)\)
\(\chi_{18975}(8989,\cdot)\)
\(\chi_{18975}(9334,\cdot)\)
\(\chi_{18975}(10204,\cdot)\)
\(\chi_{18975}(11464,\cdot)\)
\(\chi_{18975}(11809,\cdot)\)
\(\chi_{18975}(12289,\cdot)\)
\(\chi_{18975}(12394,\cdot)\)
\(\chi_{18975}(12634,\cdot)\)
\(\chi_{18975}(13219,\cdot)\)
\(\chi_{18975}(14044,\cdot)\)
...
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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 };
\((6326,10627,8626,11551)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{9}{10}\right),e\left(\frac{5}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(26\) |
| \( \chi_{ 18975 }(2734, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) |
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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)
