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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3645, base_ring=CyclotomicField(972))
M = H._module
chi = DirichletCharacter(H, M([188,729]))
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gp:[g,chi] = znchar(Mod(313, 3645))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3645.313");
| Modulus: | \(3645\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3645\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(972\) |
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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);
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\(\chi_{3645}(7,\cdot)\)
\(\chi_{3645}(13,\cdot)\)
\(\chi_{3645}(22,\cdot)\)
\(\chi_{3645}(43,\cdot)\)
\(\chi_{3645}(52,\cdot)\)
\(\chi_{3645}(58,\cdot)\)
\(\chi_{3645}(67,\cdot)\)
\(\chi_{3645}(88,\cdot)\)
\(\chi_{3645}(97,\cdot)\)
\(\chi_{3645}(103,\cdot)\)
\(\chi_{3645}(112,\cdot)\)
\(\chi_{3645}(133,\cdot)\)
\(\chi_{3645}(142,\cdot)\)
\(\chi_{3645}(148,\cdot)\)
\(\chi_{3645}(157,\cdot)\)
\(\chi_{3645}(178,\cdot)\)
\(\chi_{3645}(187,\cdot)\)
\(\chi_{3645}(193,\cdot)\)
\(\chi_{3645}(202,\cdot)\)
\(\chi_{3645}(223,\cdot)\)
\(\chi_{3645}(232,\cdot)\)
\(\chi_{3645}(238,\cdot)\)
\(\chi_{3645}(247,\cdot)\)
\(\chi_{3645}(268,\cdot)\)
\(\chi_{3645}(277,\cdot)\)
\(\chi_{3645}(283,\cdot)\)
\(\chi_{3645}(292,\cdot)\)
\(\chi_{3645}(313,\cdot)\)
\(\chi_{3645}(322,\cdot)\)
\(\chi_{3645}(328,\cdot)\)
...
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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 };
\((731,2917)\) → \((e\left(\frac{47}{243}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3645 }(313, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{917}{972}\right)\) | \(e\left(\frac{431}{486}\right)\) | \(e\left(\frac{929}{972}\right)\) | \(e\left(\frac{269}{324}\right)\) | \(e\left(\frac{179}{243}\right)\) | \(e\left(\frac{451}{972}\right)\) | \(e\left(\frac{437}{486}\right)\) | \(e\left(\frac{188}{243}\right)\) | \(e\left(\frac{43}{324}\right)\) | \(e\left(\frac{1}{162}\right)\) |
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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)
