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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3141, base_ring=CyclotomicField(348))
M = H._module
chi = DirichletCharacter(H, M([290,21]))
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gp:[g,chi] = znchar(Mod(1058, 3141))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3141.1058");
| Modulus: | \(3141\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(3141\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(348\) |
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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: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{3141}(11,\cdot)\)
\(\chi_{3141}(38,\cdot)\)
\(\chi_{3141}(47,\cdot)\)
\(\chi_{3141}(65,\cdot)\)
\(\chi_{3141}(101,\cdot)\)
\(\chi_{3141}(131,\cdot)\)
\(\chi_{3141}(146,\cdot)\)
\(\chi_{3141}(167,\cdot)\)
\(\chi_{3141}(182,\cdot)\)
\(\chi_{3141}(203,\cdot)\)
\(\chi_{3141}(218,\cdot)\)
\(\chi_{3141}(248,\cdot)\)
\(\chi_{3141}(284,\cdot)\)
\(\chi_{3141}(302,\cdot)\)
\(\chi_{3141}(311,\cdot)\)
\(\chi_{3141}(338,\cdot)\)
\(\chi_{3141}(401,\cdot)\)
\(\chi_{3141}(407,\cdot)\)
\(\chi_{3141}(410,\cdot)\)
\(\chi_{3141}(428,\cdot)\)
\(\chi_{3141}(452,\cdot)\)
\(\chi_{3141}(482,\cdot)\)
\(\chi_{3141}(536,\cdot)\)
\(\chi_{3141}(596,\cdot)\)
\(\chi_{3141}(659,\cdot)\)
\(\chi_{3141}(677,\cdot)\)
\(\chi_{3141}(704,\cdot)\)
\(\chi_{3141}(860,\cdot)\)
\(\chi_{3141}(884,\cdot)\)
\(\chi_{3141}(914,\cdot)\)
...
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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 };
\((1397,1747)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{116}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 3141 }(1058, a) \) |
\(1\) | \(1\) | \(e\left(\frac{311}{348}\right)\) | \(e\left(\frac{137}{174}\right)\) | \(e\left(\frac{10}{87}\right)\) | \(e\left(\frac{131}{348}\right)\) | \(e\left(\frac{79}{116}\right)\) | \(e\left(\frac{1}{116}\right)\) | \(e\left(\frac{35}{348}\right)\) | \(e\left(\frac{7}{348}\right)\) | \(e\left(\frac{47}{174}\right)\) | \(e\left(\frac{50}{87}\right)\) |
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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)
