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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63075, base_ring=CyclotomicField(4060))
M = H._module
chi = DirichletCharacter(H, M([2030,609,265]))
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gp:[g,chi] = znchar(Mod(533, 63075))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63075.533");
| Modulus: | \(63075\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(63075\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(4060\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{63075}(47,\cdot)\)
\(\chi_{63075}(98,\cdot)\)
\(\chi_{63075}(188,\cdot)\)
\(\chi_{63075}(242,\cdot)\)
\(\chi_{63075}(263,\cdot)\)
\(\chi_{63075}(287,\cdot)\)
\(\chi_{63075}(317,\cdot)\)
\(\chi_{63075}(338,\cdot)\)
\(\chi_{63075}(392,\cdot)\)
\(\chi_{63075}(398,\cdot)\)
\(\chi_{63075}(533,\cdot)\)
\(\chi_{63075}(617,\cdot)\)
\(\chi_{63075}(623,\cdot)\)
\(\chi_{63075}(677,\cdot)\)
\(\chi_{63075}(698,\cdot)\)
\(\chi_{63075}(722,\cdot)\)
\(\chi_{63075}(728,\cdot)\)
\(\chi_{63075}(752,\cdot)\)
\(\chi_{63075}(773,\cdot)\)
\(\chi_{63075}(833,\cdot)\)
\(\chi_{63075}(917,\cdot)\)
\(\chi_{63075}(1052,\cdot)\)
\(\chi_{63075}(1058,\cdot)\)
\(\chi_{63075}(1112,\cdot)\)
\(\chi_{63075}(1133,\cdot)\)
\(\chi_{63075}(1163,\cdot)\)
\(\chi_{63075}(1187,\cdot)\)
\(\chi_{63075}(1208,\cdot)\)
\(\chi_{63075}(1262,\cdot)\)
\(\chi_{63075}(1352,\cdot)\)
...
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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 };
\((21026,30277,11776)\) → \((-1,e\left(\frac{3}{20}\right),e\left(\frac{53}{812}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 63075 }(533, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{726}{1015}\right)\) | \(e\left(\frac{437}{1015}\right)\) | \(e\left(\frac{237}{812}\right)\) | \(e\left(\frac{148}{1015}\right)\) | \(e\left(\frac{759}{4060}\right)\) | \(e\left(\frac{241}{4060}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{874}{1015}\right)\) | \(e\left(\frac{9}{145}\right)\) | \(e\left(\frac{2007}{4060}\right)\) |
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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)
