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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21850, base_ring=CyclotomicField(1980))
M = H._module
chi = DirichletCharacter(H, M([1089,880,1530]))
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gp:[g,chi] = znchar(Mod(1073, 21850))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21850.1073");
| Modulus: | \(21850\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10925\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(1980\) |
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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 \(\chi_{10925}(1073,\cdot)\) |
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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);
|
\(\chi_{21850}(17,\cdot)\)
\(\chi_{21850}(63,\cdot)\)
\(\chi_{21850}(237,\cdot)\)
\(\chi_{21850}(263,\cdot)\)
\(\chi_{21850}(283,\cdot)\)
\(\chi_{21850}(313,\cdot)\)
\(\chi_{21850}(327,\cdot)\)
\(\chi_{21850}(503,\cdot)\)
\(\chi_{21850}(517,\cdot)\)
\(\chi_{21850}(567,\cdot)\)
\(\chi_{21850}(613,\cdot)\)
\(\chi_{21850}(617,\cdot)\)
\(\chi_{21850}(663,\cdot)\)
\(\chi_{21850}(727,\cdot)\)
\(\chi_{21850}(747,\cdot)\)
\(\chi_{21850}(803,\cdot)\)
\(\chi_{21850}(833,\cdot)\)
\(\chi_{21850}(917,\cdot)\)
\(\chi_{21850}(937,\cdot)\)
\(\chi_{21850}(973,\cdot)\)
\(\chi_{21850}(1023,\cdot)\)
\(\chi_{21850}(1073,\cdot)\)
\(\chi_{21850}(1137,\cdot)\)
\(\chi_{21850}(1183,\cdot)\)
\(\chi_{21850}(1187,\cdot)\)
\(\chi_{21850}(1203,\cdot)\)
\(\chi_{21850}(1213,\cdot)\)
\(\chi_{21850}(1233,\cdot)\)
\(\chi_{21850}(1263,\cdot)\)
\(\chi_{21850}(1353,\cdot)\)
...
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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 };
\((20977,2301,16151)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{4}{9}\right),e\left(\frac{17}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 21850 }(1073, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1963}{1980}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{973}{990}\right)\) | \(e\left(\frac{29}{330}\right)\) | \(e\left(\frac{971}{1980}\right)\) | \(e\left(\frac{7}{1980}\right)\) | \(e\left(\frac{89}{990}\right)\) | \(e\left(\frac{643}{660}\right)\) | \(e\left(\frac{559}{990}\right)\) | \(e\left(\frac{116}{165}\right)\) |
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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)
