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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(296205, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([210,315,80,280,168]))
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gp:[g,chi] = znchar(Mod(80453, 296205))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("296205.80453");
| Modulus: | \(296205\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(296205\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(420\) |
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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);
|
\(\chi_{296205}(1628,\cdot)\)
\(\chi_{296205}(10637,\cdot)\)
\(\chi_{296205}(13022,\cdot)\)
\(\chi_{296205}(15752,\cdot)\)
\(\chi_{296205}(16097,\cdot)\)
\(\chi_{296205}(21212,\cdot)\)
\(\chi_{296205}(24833,\cdot)\)
\(\chi_{296205}(27563,\cdot)\)
\(\chi_{296205}(29948,\cdot)\)
\(\chi_{296205}(32132,\cdot)\)
\(\chi_{296205}(32678,\cdot)\)
\(\chi_{296205}(33023,\cdot)\)
\(\chi_{296205}(38138,\cdot)\)
\(\chi_{296205}(43943,\cdot)\)
\(\chi_{296205}(49058,\cdot)\)
\(\chi_{296205}(50222,\cdot)\)
\(\chi_{296205}(52952,\cdot)\)
\(\chi_{296205}(55337,\cdot)\)
\(\chi_{296205}(58067,\cdot)\)
\(\chi_{296205}(58412,\cdot)\)
\(\chi_{296205}(63527,\cdot)\)
\(\chi_{296205}(69332,\cdot)\)
\(\chi_{296205}(69878,\cdot)\)
\(\chi_{296205}(72263,\cdot)\)
\(\chi_{296205}(74447,\cdot)\)
\(\chi_{296205}(74993,\cdot)\)
\(\chi_{296205}(75338,\cdot)\)
\(\chi_{296205}(80453,\cdot)\)
\(\chi_{296205}(91373,\cdot)\)
\(\chi_{296205}(92537,\cdot)\)
...
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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 };
\((197471,59242,66496,68356,171991)\) → \((-1,-i,e\left(\frac{4}{21}\right),e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) | \(29\) |
| \( \chi_{ 296205 }(80453, a) \) |
\(1\) | \(1\) | \(e\left(\frac{197}{420}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{73}{105}\right)\) |
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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)
