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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1769, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([195,56]))
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gp:[g,chi] = znchar(Mod(22, 1769))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1769.22");
| Modulus: | \(1769\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(1769\) |
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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: | even |
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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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\(\chi_{1769}(22,\cdot)\)
\(\chi_{1769}(42,\cdot)\)
\(\chi_{1769}(138,\cdot)\)
\(\chi_{1769}(178,\cdot)\)
\(\chi_{1769}(179,\cdot)\)
\(\chi_{1769}(208,\cdot)\)
\(\chi_{1769}(225,\cdot)\)
\(\chi_{1769}(266,\cdot)\)
\(\chi_{1769}(361,\cdot)\)
\(\chi_{1769}(381,\cdot)\)
\(\chi_{1769}(382,\cdot)\)
\(\chi_{1769}(439,\cdot)\)
\(\chi_{1769}(469,\cdot)\)
\(\chi_{1769}(544,\cdot)\)
\(\chi_{1769}(564,\cdot)\)
\(\chi_{1769}(622,\cdot)\)
\(\chi_{1769}(747,\cdot)\)
\(\chi_{1769}(788,\cdot)\)
\(\chi_{1769}(789,\cdot)\)
\(\chi_{1769}(805,\cdot)\)
\(\chi_{1769}(818,\cdot)\)
\(\chi_{1769}(850,\cdot)\)
\(\chi_{1769}(876,\cdot)\)
\(\chi_{1769}(879,\cdot)\)
\(\chi_{1769}(937,\cdot)\)
\(\chi_{1769}(991,\cdot)\)
\(\chi_{1769}(992,\cdot)\)
\(\chi_{1769}(1049,\cdot)\)
\(\chi_{1769}(1053,\cdot)\)
\(\chi_{1769}(1079,\cdot)\)
...
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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 };
\((611,1161)\) → \((e\left(\frac{13}{14}\right),e\left(\frac{4}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1769 }(22, a) \) |
\(1\) | \(1\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{46}{105}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{3}{14}\right)\) |
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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)
