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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2596, base_ring=CyclotomicField(290))
M = H._module
chi = DirichletCharacter(H, M([145,87,215]))
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gp:[g,chi] = znchar(Mod(195, 2596))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2596.195");
| Modulus: | \(2596\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(2596\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(290\) |
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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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\(\chi_{2596}(39,\cdot)\)
\(\chi_{2596}(83,\cdot)\)
\(\chi_{2596}(151,\cdot)\)
\(\chi_{2596}(183,\cdot)\)
\(\chi_{2596}(195,\cdot)\)
\(\chi_{2596}(211,\cdot)\)
\(\chi_{2596}(215,\cdot)\)
\(\chi_{2596}(227,\cdot)\)
\(\chi_{2596}(259,\cdot)\)
\(\chi_{2596}(283,\cdot)\)
\(\chi_{2596}(303,\cdot)\)
\(\chi_{2596}(327,\cdot)\)
\(\chi_{2596}(347,\cdot)\)
\(\chi_{2596}(387,\cdot)\)
\(\chi_{2596}(391,\cdot)\)
\(\chi_{2596}(415,\cdot)\)
\(\chi_{2596}(431,\cdot)\)
\(\chi_{2596}(447,\cdot)\)
\(\chi_{2596}(503,\cdot)\)
\(\chi_{2596}(519,\cdot)\)
\(\chi_{2596}(563,\cdot)\)
\(\chi_{2596}(623,\cdot)\)
\(\chi_{2596}(651,\cdot)\)
\(\chi_{2596}(655,\cdot)\)
\(\chi_{2596}(667,\cdot)\)
\(\chi_{2596}(679,\cdot)\)
\(\chi_{2596}(699,\cdot)\)
\(\chi_{2596}(739,\cdot)\)
\(\chi_{2596}(755,\cdot)\)
\(\chi_{2596}(799,\cdot)\)
...
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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 };
\((1299,2125,2421)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{43}{58}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 2596 }(195, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{281}{290}\right)\) | \(e\left(\frac{94}{145}\right)\) | \(e\left(\frac{137}{145}\right)\) | \(e\left(\frac{136}{145}\right)\) | \(e\left(\frac{96}{145}\right)\) | \(e\left(\frac{179}{290}\right)\) | \(e\left(\frac{103}{290}\right)\) | \(e\left(\frac{83}{145}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) |
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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)
