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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10153, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([126,35,51]))
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gp:[g,chi] = znchar(Mod(3683, 10153))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10153.3683");
| Modulus: | \(10153\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(10153\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(210\) |
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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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\(\chi_{10153}(69,\cdot)\)
\(\chi_{10153}(621,\cdot)\)
\(\chi_{10153}(686,\cdot)\)
\(\chi_{10153}(1291,\cdot)\)
\(\chi_{10153}(1356,\cdot)\)
\(\chi_{10153}(1479,\cdot)\)
\(\chi_{10153}(1830,\cdot)\)
\(\chi_{10153}(1928,\cdot)\)
\(\chi_{10153}(1973,\cdot)\)
\(\chi_{10153}(2051,\cdot)\)
\(\chi_{10153}(2324,\cdot)\)
\(\chi_{10153}(2506,\cdot)\)
\(\chi_{10153}(2513,\cdot)\)
\(\chi_{10153}(2649,\cdot)\)
\(\chi_{10153}(2688,\cdot)\)
\(\chi_{10153}(3683,\cdot)\)
\(\chi_{10153}(4112,\cdot)\)
\(\chi_{10153}(4222,\cdot)\)
\(\chi_{10153}(4398,\cdot)\)
\(\chi_{10153}(4437,\cdot)\)
\(\chi_{10153}(4515,\cdot)\)
\(\chi_{10153}(4612,\cdot)\)
\(\chi_{10153}(4755,\cdot)\)
\(\chi_{10153}(4788,\cdot)\)
\(\chi_{10153}(4801,\cdot)\)
\(\chi_{10153}(5977,\cdot)\)
\(\chi_{10153}(6042,\cdot)\)
\(\chi_{10153}(6088,\cdot)\)
\(\chi_{10153}(6153,\cdot)\)
\(\chi_{10153}(6614,\cdot)\)
...
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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 };
\((9231,782,859)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{1}{6}\right),e\left(\frac{17}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 10153 }(3683, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{8}{35}\right)\) |
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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)
