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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1769, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([375,224]))
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gp:[g,chi] = znchar(Mod(301, 1769))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1769.301");
| 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: | \(420\) |
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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: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
|
\(\chi_{1769}(15,\cdot)\)
\(\chi_{1769}(56,\cdot)\)
\(\chi_{1769}(73,\cdot)\)
\(\chi_{1769}(76,\cdot)\)
\(\chi_{1769}(77,\cdot)\)
\(\chi_{1769}(118,\cdot)\)
\(\chi_{1769}(134,\cdot)\)
\(\chi_{1769}(137,\cdot)\)
\(\chi_{1769}(147,\cdot)\)
\(\chi_{1769}(164,\cdot)\)
\(\chi_{1769}(195,\cdot)\)
\(\chi_{1769}(205,\cdot)\)
\(\chi_{1769}(240,\cdot)\)
\(\chi_{1769}(259,\cdot)\)
\(\chi_{1769}(269,\cdot)\)
\(\chi_{1769}(300,\cdot)\)
\(\chi_{1769}(301,\cdot)\)
\(\chi_{1769}(317,\cdot)\)
\(\chi_{1769}(321,\cdot)\)
\(\chi_{1769}(327,\cdot)\)
\(\chi_{1769}(330,\cdot)\)
\(\chi_{1769}(362,\cdot)\)
\(\chi_{1769}(388,\cdot)\)
\(\chi_{1769}(391,\cdot)\)
\(\chi_{1769}(408,\cdot)\)
\(\chi_{1769}(443,\cdot)\)
\(\chi_{1769}(449,\cdot)\)
\(\chi_{1769}(483,\cdot)\)
\(\chi_{1769}(503,\cdot)\)
\(\chi_{1769}(504,\cdot)\)
...
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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 };
\((611,1161)\) → \((e\left(\frac{25}{28}\right),e\left(\frac{8}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1769 }(301, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{179}{420}\right)\) | \(e\left(\frac{93}{140}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{9}{28}\right)\) |
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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)
